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Theorem rlocaddval 33226
Description: Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
rlocaddval.1 𝐵 = (Base‘𝑅)
rlocaddval.2 · = (.r‘𝑅)
rlocaddval.3 + = (+g‘𝑅)
rlocaddval.4 𝐿 = (𝑅 RLocal 𝑆)
rlocaddval.5 ∼ = (𝑅 ~RL 𝑆)
rlocaddval.r (𝜑 → 𝑅 ∈ CRing)
rlocaddval.s (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
rlocaddval.6 (𝜑 → 𝐸 ∈ 𝐵)
rlocaddval.7 (𝜑 → 𝐹 ∈ 𝐵)
rlocaddval.8 (𝜑 → 𝐺 ∈ 𝑆)
rlocaddval.9 (𝜑 → 𝐻 ∈ 𝑆)
rlocaddval.10 ⊕ = (+g‘𝐿)
Assertion
Ref Expression
rlocaddval (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] ∼ )
Proof of Theorem rlocaddval
Dummy variables 𝑝 𝑞 𝑢 𝑣 𝑎 𝑏 𝑓 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlocaddval.6 . . . 4 (𝜑 → 𝐸 ∈ 𝐵)
2 rlocaddval.8 . . . 4 (𝜑 → 𝐺 ∈ 𝑆)
31, 2opelxpd 5680 . . 3 (𝜑 → ⟨𝐸, 𝐺⟩ ∈ (𝐵 × 𝑆))
4 rlocaddval.7 . . . 4 (𝜑 → 𝐹 ∈ 𝐵)
5 rlocaddval.9 . . . 4 (𝜑 → 𝐻 ∈ 𝑆)
64, 5opelxpd 5680 . . 3 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝐵 × 𝑆))
7 rlocaddval.4 . . . . 5 𝐿 = (𝑅 RLocal 𝑆)
8 rlocaddval.1 . . . . . 6 𝐵 = (Base‘𝑅)
9 eqid 2730 . . . . . 6 (0g‘𝑅) = (0g‘𝑅)
10 rlocaddval.2 . . . . . 6 · = (.r‘𝑅)
11 eqid 2730 . . . . . 6 (-g‘𝑅) = (-g‘𝑅)
12 rlocaddval.3 . . . . . 6 + = (+g‘𝑅)
13 eqid 2730 . . . . . 6 (le‘𝑅) = (le‘𝑅)
14 eqid 2730 . . . . . 6 (Scalar‘𝑅) = (Scalar‘𝑅)
15 eqid 2730 . . . . . 6 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
16 eqid 2730 . . . . . 6 ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
17 eqid 2730 . . . . . 6 (𝐵 × 𝑆) = (𝐵 × 𝑆)
18 rlocaddval.5 . . . . . 6 ∼ = (𝑅 ~RL 𝑆)
19 eqid 2730 . . . . . 6 (TopSet‘𝑅) = (TopSet‘𝑅)
20 eqid 2730 . . . . . 6 (dist‘𝑅) = (dist‘𝑅)
21 eqid 2730 . . . . . 6 (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) = (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)
22 eqid 2730 . . . . . 6 (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) = (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)
23 eqid 2730 . . . . . 6 (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩) = (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)
24 eqid 2730 . . . . . 6 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}
25 eqid 2730 . . . . . 6 (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))
26 rlocaddval.r . . . . . 6 (𝜑 → 𝑅 ∈ CRing)
27 rlocaddval.s . . . . . . 7 (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
28 eqid 2730 . . . . . . . . 9 (mulGrp‘𝑅) = (mulGrp‘𝑅)
2928, 8mgpbas 20061 . . . . . . . 8 𝐵 = (Base‘(mulGrp‘𝑅))
3029submss 18743 . . . . . . 7 (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
3127, 30syl 17 . . . . . 6 (𝜑 → 𝑆 ⊆ 𝐵)
328, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 31rlocval 33217 . . . . 5 (𝜑 → (𝑅 RLocal 𝑆) = ((({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}) /s ∼ ))
337, 32eqtrid 2777 . . . 4 (𝜑 → 𝐿 = ((({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}) /s ∼ ))
34 eqidd 2731 . . . . . 6 (𝜑 → (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}) = (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))
35 eqid 2730 . . . . . . 7 (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}) = (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})
3635imasvalstr 17421 . . . . . 6 (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}) Struct ⟨1, 12⟩
37 baseid 17189 . . . . . 6 Base = Slot (Base‘ndx)
38 snsstp1 4783 . . . . . . 7 {⟨(Base‘ndx), (𝐵 × 𝑆)⟩} ⊆ {⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩}
39 ssun1 4144 . . . . . . . 8 {⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ⊆ ({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩})
40 ssun1 4144 . . . . . . . 8 ({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ⊆ (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})
4139, 40sstri 3959 . . . . . . 7 {⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ⊆ (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})
4238, 41sstri 3959 . . . . . 6 {⟨(Base‘ndx), (𝐵 × 𝑆)⟩} ⊆ (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})
438fvexi 6875 . . . . . . . 8 𝐵 ∈ V
4443a1i 11 . . . . . . 7 (𝜑 → 𝐵 ∈ V)
4544, 27xpexd 7730 . . . . . 6 (𝜑 → (𝐵 × 𝑆) ∈ V)
46 eqid 2730 . . . . . 6 (Base‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})) = (Base‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))
4734, 36, 37, 42, 45, 46strfv3 17181 . . . . 5 (𝜑 → (Base‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})) = (𝐵 × 𝑆))
4847eqcomd 2736 . . . 4 (𝜑 → (𝐵 × 𝑆) = (Base‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})))
49 eqid 2730 . . . . 5 (1r‘𝑅) = (1r‘𝑅)
508, 9, 49, 10, 11, 17, 18, 26, 27erler 33223 . . . 4 (𝜑 → ∼ Er (𝐵 × 𝑆))
51 tpex 7725 . . . . . . 7 {⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∈ V
52 tpex 7725 . . . . . . 7 {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩} ∈ V
5351, 52unex 7723 . . . . . 6 ({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∈ V
54 tpex 7725 . . . . . 6 {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩} ∈ V
5553, 54unex 7723 . . . . 5 (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}) ∈ V
5655a1i 11 . . . 4 (𝜑 → (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}) ∈ V)
5731ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑆 ⊆ 𝐵)
5857ad2antrr 726 . . . . . . . . . . 11 (((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) → 𝑆 ⊆ 𝐵)
5958ad2antrr 726 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑆 ⊆ 𝐵)
60 eqidd 2731 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ = ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩)
61 eqidd 2731 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩ = ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
6226crnggrpd 20163 . . . . . . . . . . . 12 (𝜑 → 𝑅 ∈ Grp)
6362ad6antr 736 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑅 ∈ Grp)
6426crngringd 20162 . . . . . . . . . . . . 13 (𝜑 → 𝑅 ∈ Ring)
6564ad6antr 736 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑅 ∈ Ring)
66 simplr 768 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑢 ∼ 𝑝)
678, 18, 57, 66erlcl1 33218 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑢 ∈ (𝐵 × 𝑆))
6867ad4antr 732 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑢 ∈ (𝐵 × 𝑆))
69 xp1st 8003 . . . . . . . . . . . . 13 (𝑢 ∈ (𝐵 × 𝑆) → (1st ‘𝑢) ∈ 𝐵)
7068, 69syl 17 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (1st ‘𝑢) ∈ 𝐵)
71 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑣 ∼ 𝑞)
728, 18, 57, 71erlcl1 33218 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑣 ∈ (𝐵 × 𝑆))
7372ad4antr 732 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑣 ∈ (𝐵 × 𝑆))
74 xp2nd 8004 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝐵 × 𝑆) → (2nd ‘𝑣) ∈ 𝑆)
7573, 74syl 17 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑣) ∈ 𝑆)
7659, 75sseldd 3950 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑣) ∈ 𝐵)
778, 10, 65, 70, 76ringcld 20176 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑢) · (2nd ‘𝑣)) ∈ 𝐵)
78 xp1st 8003 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐵 × 𝑆) → (1st ‘𝑣) ∈ 𝐵)
7973, 78syl 17 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (1st ‘𝑣) ∈ 𝐵)
80 xp2nd 8004 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐵 × 𝑆) → (2nd ‘𝑢) ∈ 𝑆)
8168, 80syl 17 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑢) ∈ 𝑆)
8259, 81sseldd 3950 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑢) ∈ 𝐵)
838, 10, 65, 79, 82ringcld 20176 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑣) · (2nd ‘𝑢)) ∈ 𝐵)
848, 12, 63, 77, 83grpcld 18886 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))) ∈ 𝐵)
858, 18, 57, 66erlcl2 33219 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑝 ∈ (𝐵 × 𝑆))
8685ad4antr 732 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑝 ∈ (𝐵 × 𝑆))
87 xp1st 8003 . . . . . . . . . . . . 13 (𝑝 ∈ (𝐵 × 𝑆) → (1st ‘𝑝) ∈ 𝐵)
8886, 87syl 17 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (1st ‘𝑝) ∈ 𝐵)
898, 18, 57, 71erlcl2 33219 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑞 ∈ (𝐵 × 𝑆))
9089ad4antr 732 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑞 ∈ (𝐵 × 𝑆))
91 xp2nd 8004 . . . . . . . . . . . . . 14 (𝑞 ∈ (𝐵 × 𝑆) → (2nd ‘𝑞) ∈ 𝑆)
9290, 91syl 17 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑞) ∈ 𝑆)
9359, 92sseldd 3950 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑞) ∈ 𝐵)
948, 10, 65, 88, 93ringcld 20176 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑝) · (2nd ‘𝑞)) ∈ 𝐵)
95 xp1st 8003 . . . . . . . . . . . . 13 (𝑞 ∈ (𝐵 × 𝑆) → (1st ‘𝑞) ∈ 𝐵)
9690, 95syl 17 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (1st ‘𝑞) ∈ 𝐵)
97 xp2nd 8004 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝐵 × 𝑆) → (2nd ‘𝑝) ∈ 𝑆)
9886, 97syl 17 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑝) ∈ 𝑆)
9959, 98sseldd 3950 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (2nd ‘𝑝) ∈ 𝐵)
1008, 10, 65, 96, 99ringcld 20176 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑞) · (2nd ‘𝑝)) ∈ 𝐵)
1018, 12, 63, 94, 100grpcld 18886 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) ∈ 𝐵)
10227ad6antr 736 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
10328, 10mgpplusg 20060 . . . . . . . . . . . 12 · = (+g‘(mulGrp‘𝑅))
104103submcl 18746 . . . . . . . . . . 11 ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ (2nd ‘𝑢) ∈ 𝑆 ∧ (2nd ‘𝑣) ∈ 𝑆) → ((2nd ‘𝑢) · (2nd ‘𝑣)) ∈ 𝑆)
105102, 81, 75, 104syl3anc 1373 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑢) · (2nd ‘𝑣)) ∈ 𝑆)
106103submcl 18746 . . . . . . . . . . 11 ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ (2nd ‘𝑝) ∈ 𝑆 ∧ (2nd ‘𝑞) ∈ 𝑆) → ((2nd ‘𝑝) · (2nd ‘𝑞)) ∈ 𝑆)
107102, 98, 92, 106syl3anc 1373 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑝) · (2nd ‘𝑞)) ∈ 𝑆)
108 simp-4r 783 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑓 ∈ 𝑆)
109 simplr 768 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑔 ∈ 𝑆)
110103submcl 18746 . . . . . . . . . . 11 ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ 𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) → (𝑓 · 𝑔) ∈ 𝑆)
111102, 108, 109, 110syl3anc 1373 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · 𝑔) ∈ 𝑆)
11259, 107sseldd 3950 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑝) · (2nd ‘𝑞)) ∈ 𝐵)
1138, 12, 10ringdir 20178 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (((1st ‘𝑢) · (2nd ‘𝑣)) ∈ 𝐵 ∧ ((1st ‘𝑣) · (2nd ‘𝑢)) ∈ 𝐵 ∧ ((2nd ‘𝑝) · (2nd ‘𝑞)) ∈ 𝐵)) → ((((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))) · ((2nd ‘𝑝) · (2nd ‘𝑞))) = ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))
11465, 77, 83, 112, 113syl13anc 1374 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))) · ((2nd ‘𝑝) · (2nd ‘𝑞))) = ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))
11559, 105sseldd 3950 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑢) · (2nd ‘𝑣)) ∈ 𝐵)
1168, 12, 10ringdir 20178 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (((1st ‘𝑝) · (2nd ‘𝑞)) ∈ 𝐵 ∧ ((1st ‘𝑞) · (2nd ‘𝑝)) ∈ 𝐵 ∧ ((2nd ‘𝑢) · (2nd ‘𝑣)) ∈ 𝐵)) → ((((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) · ((2nd ‘𝑢) · (2nd ‘𝑣))) = ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))
11765, 94, 100, 115, 116syl13anc 1374 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) · ((2nd ‘𝑢) · (2nd ‘𝑣))) = ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))
118114, 117oveq12d 7408 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))) · ((2nd ‘𝑝) · (2nd ‘𝑞)))(-g‘𝑅)((((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) = (((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))))
119118oveq2d 7406 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))) · ((2nd ‘𝑝) · (2nd ‘𝑞)))(-g‘𝑅)((((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = ((𝑓 · 𝑔) · (((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))))
12059, 108sseldd 3950 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑓 ∈ 𝐵)
12159, 109sseldd 3950 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑔 ∈ 𝐵)
1228, 10, 65, 120, 121ringcld 20176 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · 𝑔) ∈ 𝐵)
1238, 10, 65, 77, 112ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) ∈ 𝐵)
1248, 10, 65, 83, 112ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) ∈ 𝐵)
1258, 12, 63, 123, 124grpcld 18886 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) ∈ 𝐵)
1268, 10, 65, 94, 115ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) ∈ 𝐵)
1278, 10, 65, 100, 115ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) ∈ 𝐵)
1288, 12, 63, 126, 127grpcld 18886 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) ∈ 𝐵)
1298, 10, 11, 65, 122, 125, 128ringsubdi 20223 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))) = (((𝑓 · 𝑔) · ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))(-g‘𝑅)((𝑓 · 𝑔) · ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))))
1308, 12, 10ringdi 20177 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ ((𝑓 · 𝑔) ∈ 𝐵 ∧ (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) ∈ 𝐵 ∧ (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) ∈ 𝐵)) → ((𝑓 · 𝑔) · ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))) = (((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) + ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))))
13165, 122, 123, 124, 130syl13anc 1374 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))) = (((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) + ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))))
1328, 12, 10ringdi 20177 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ ((𝑓 · 𝑔) ∈ 𝐵 ∧ (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) ∈ 𝐵 ∧ (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) ∈ 𝐵)) → ((𝑓 · 𝑔) · ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = (((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) + ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))))
13365, 122, 126, 127, 132syl13anc 1374 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = (((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) + ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))))
134131, 133oveq12d 7408 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑓 · 𝑔) · ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))(-g‘𝑅)((𝑓 · 𝑔) · ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))) = ((((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) + ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))(-g‘𝑅)(((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) + ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))))
13565ringabld 20199 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → 𝑅 ∈ Abel)
1368, 10, 65, 122, 123ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) ∈ 𝐵)
1378, 10, 65, 122, 124ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) ∈ 𝐵)
1388, 10, 65, 122, 126ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) ∈ 𝐵)
1398, 10, 65, 122, 127ringcld 20176 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) ∈ 𝐵)
1408, 12, 11ablsub4 19747 . . . . . . . . . . . . 13 ((𝑅 ∈ Abel ∧ (((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) ∈ 𝐵 ∧ ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) ∈ 𝐵) ∧ (((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) ∈ 𝐵 ∧ ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) ∈ 𝐵)) → ((((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) + ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))(-g‘𝑅)(((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) + ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))) = ((((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) + (((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))))
141135, 136, 137, 138, 139, 140syl122anc 1381 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) + ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))(-g‘𝑅)(((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) + ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))) = ((((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) + (((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))))
14228crngmgp 20157 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
14326, 142syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
144143ad6antr 736 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (mulGrp‘𝑅) ∈ CMnd)
14529, 103, 144, 120, 121, 70, 76, 99, 93cmn246135 32981 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) = ((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝)))))
14629, 103, 144, 120, 121, 88, 93, 82, 76cmn246135 32981 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) = ((𝑔 · ((2nd ‘𝑞) · (2nd ‘𝑣))) · (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢)))))
14729, 103cmncom 19735 . . . . . . . . . . . . . . . . . . . 20 (((mulGrp‘𝑅) ∈ CMnd ∧ (2nd ‘𝑣) ∈ 𝐵 ∧ (2nd ‘𝑞) ∈ 𝐵) → ((2nd ‘𝑣) · (2nd ‘𝑞)) = ((2nd ‘𝑞) · (2nd ‘𝑣)))
148144, 76, 93, 147syl3anc 1373 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑣) · (2nd ‘𝑞)) = ((2nd ‘𝑞) · (2nd ‘𝑣)))
149148oveq2d 7406 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) = (𝑔 · ((2nd ‘𝑞) · (2nd ‘𝑣))))
150149oveq1d 7405 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢)))) = ((𝑔 · ((2nd ‘𝑞) · (2nd ‘𝑣))) · (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢)))))
151146, 150eqtr4d 2768 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) = ((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢)))))
152145, 151oveq12d 7408 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = (((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝))))(-g‘𝑅)((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢))))))
1538, 10, 65, 70, 99ringcld 20176 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑢) · (2nd ‘𝑝)) ∈ 𝐵)
1548, 10, 65, 88, 82ringcld 20176 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑝) · (2nd ‘𝑢)) ∈ 𝐵)
1558, 10, 11, 65, 120, 153, 154ringsubdi 20223 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = ((𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝)))(-g‘𝑅)(𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢)))))
156 simpllr 775 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅))
157155, 156eqtr3d 2767 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝)))(-g‘𝑅)(𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅))
158157oveq2d 7406 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · ((𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝)))(-g‘𝑅)(𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢))))) = ((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (0g‘𝑅)))
1598, 10, 65, 76, 93ringcld 20176 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑣) · (2nd ‘𝑞)) ∈ 𝐵)
1608, 10, 65, 121, 159ringcld 20176 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) ∈ 𝐵)
1618, 10, 65, 120, 153ringcld 20176 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝))) ∈ 𝐵)
1628, 10, 65, 120, 154ringcld 20176 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢))) ∈ 𝐵)
1638, 10, 11, 65, 160, 161, 162ringsubdi 20223 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · ((𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝)))(-g‘𝑅)(𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢))))) = (((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝))))(-g‘𝑅)((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢))))))
1648, 10, 9, 65, 160ringrzd 20212 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (0g‘𝑅)) = (0g‘𝑅))
165158, 163, 1643eqtr3d 2773 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑢) · (2nd ‘𝑝))))(-g‘𝑅)((𝑔 · ((2nd ‘𝑣) · (2nd ‘𝑞))) · (𝑓 · ((1st ‘𝑝) · (2nd ‘𝑢))))) = (0g‘𝑅))
166152, 165eqtrd 2765 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = (0g‘𝑅))
16729, 103, 144, 120, 121, 79, 82, 99, 93cmn145236 32982 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))) = ((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞)))))
16829, 103, 144, 120, 121, 96, 99, 82, 76cmn145236 32982 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) = ((𝑓 · ((2nd ‘𝑝) · (2nd ‘𝑢))) · (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣)))))
16929, 103cmncom 19735 . . . . . . . . . . . . . . . . . . . 20 (((mulGrp‘𝑅) ∈ CMnd ∧ (2nd ‘𝑝) ∈ 𝐵 ∧ (2nd ‘𝑢) ∈ 𝐵) → ((2nd ‘𝑝) · (2nd ‘𝑢)) = ((2nd ‘𝑢) · (2nd ‘𝑝)))
170144, 99, 82, 169syl3anc 1373 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑝) · (2nd ‘𝑢)) = ((2nd ‘𝑢) · (2nd ‘𝑝)))
171170oveq2d 7406 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · ((2nd ‘𝑝) · (2nd ‘𝑢))) = (𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))))
172171oveq1d 7405 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · ((2nd ‘𝑝) · (2nd ‘𝑢))) · (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣)))) = ((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣)))))
173168, 172eqtrd 2765 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))) = ((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣)))))
174167, 173oveq12d 7408 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = (((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣))))))
1758, 10, 65, 79, 93ringcld 20176 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑣) · (2nd ‘𝑞)) ∈ 𝐵)
1768, 10, 65, 96, 76ringcld 20176 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((1st ‘𝑞) · (2nd ‘𝑣)) ∈ 𝐵)
1778, 10, 11, 65, 121, 175, 176ringsubdi 20223 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = ((𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞)))(-g‘𝑅)(𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣)))))
178 simpr 484 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅))
179177, 178eqtr3d 2767 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞)))(-g‘𝑅)(𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅))
180179oveq2d 7406 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · ((𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞)))(-g‘𝑅)(𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣))))) = ((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (0g‘𝑅)))
1818, 10, 65, 82, 99ringcld 20176 . . . . . . . . . . . . . . . . . 18 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((2nd ‘𝑢) · (2nd ‘𝑝)) ∈ 𝐵)
1828, 10, 65, 120, 181ringcld 20176 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) ∈ 𝐵)
1838, 10, 65, 121, 175ringcld 20176 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞))) ∈ 𝐵)
1848, 10, 65, 121, 176ringcld 20176 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣))) ∈ 𝐵)
1858, 10, 11, 65, 182, 183, 184ringsubdi 20223 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · ((𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞)))(-g‘𝑅)(𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣))))) = (((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣))))))
1868, 10, 9, 65, 182ringrzd 20212 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (0g‘𝑅)) = (0g‘𝑅))
187180, 185, 1863eqtr3d 2773 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑣) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · ((2nd ‘𝑢) · (2nd ‘𝑝))) · (𝑔 · ((1st ‘𝑞) · (2nd ‘𝑣))))) = (0g‘𝑅))
188174, 187eqtrd 2765 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = (0g‘𝑅))
189166, 188oveq12d 7408 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) + (((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))) = ((0g‘𝑅) + (0g‘𝑅)))
1908, 9grpidcl 18904 . . . . . . . . . . . . . . 15 (𝑅 ∈ Grp → (0g‘𝑅) ∈ 𝐵)
19163, 190syl 17 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (0g‘𝑅) ∈ 𝐵)
1928, 12, 9, 63, 191grplidd 18908 . . . . . . . . . . . . 13 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((0g‘𝑅) + (0g‘𝑅)) = (0g‘𝑅))
193189, 192eqtrd 2765 . . . . . . . . . . . 12 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((((𝑓 · 𝑔) · (((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) + (((𝑓 · 𝑔) · (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞))))(-g‘𝑅)((𝑓 · 𝑔) · (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))) = (0g‘𝑅))
194134, 141, 1933eqtrd 2769 . . . . . . . . . . 11 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → (((𝑓 · 𝑔) · ((((1st ‘𝑢) · (2nd ‘𝑣)) · ((2nd ‘𝑝) · (2nd ‘𝑞))) + (((1st ‘𝑣) · (2nd ‘𝑢)) · ((2nd ‘𝑝) · (2nd ‘𝑞)))))(-g‘𝑅)((𝑓 · 𝑔) · ((((1st ‘𝑝) · (2nd ‘𝑞)) · ((2nd ‘𝑢) · (2nd ‘𝑣))) + (((1st ‘𝑞) · (2nd ‘𝑝)) · ((2nd ‘𝑢) · (2nd ‘𝑣)))))) = (0g‘𝑅))
195119, 129, 1943eqtrd 2769 . . . . . . . . . 10 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ((𝑓 · 𝑔) · (((((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))) · ((2nd ‘𝑝) · (2nd ‘𝑞)))(-g‘𝑅)((((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) · ((2nd ‘𝑢) · (2nd ‘𝑣))))) = (0g‘𝑅))
1968, 18, 59, 9, 10, 11, 60, 61, 84, 101, 105, 107, 111, 195erlbrd 33221 . . . . . . . . 9 (((((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) ∧ 𝑔 ∈ 𝑆) ∧ (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅)) → ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ ∼ ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
19771ad2antrr 726 . . . . . . . . . 10 (((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) → 𝑣 ∼ 𝑞)
1988, 18, 58, 9, 10, 11, 197erldi 33220 . . . . . . . . 9 (((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) → ∃𝑔 ∈ 𝑆 (𝑔 · (((1st ‘𝑣) · (2nd ‘𝑞))(-g‘𝑅)((1st ‘𝑞) · (2nd ‘𝑣)))) = (0g‘𝑅))
199196, 198r19.29a 3142 . . . . . . . 8 (((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) → ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ ∼ ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
2008, 18, 57, 9, 10, 11, 66erldi 33220 . . . . . . . 8 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → ∃𝑓 ∈ 𝑆 (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅))
201199, 200r19.29a 3142 . . . . . . 7 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ ∼ ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
202 plusgid 17254 . . . . . . . . . . . 12 +g = Slot (+g‘ndx)
203 snsstp2 4784 . . . . . . . . . . . . 13 {⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ⊆ {⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩}
204203, 41sstri 3959 . . . . . . . . . . . 12 {⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ⊆ (({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})
20521mpoexg 8058 . . . . . . . . . . . . 13 (((𝐵 × 𝑆) ∈ V ∧ (𝐵 × 𝑆) ∈ V) → (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) ∈ V)
20645, 45, 205syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) ∈ V)
207 eqid 2730 . . . . . . . . . . . 12 (+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})) = (+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))
20834, 36, 202, 204, 206, 207strfv3 17181 . . . . . . . . . . 11 (𝜑 → (+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})) = (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩))
209208ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩})) = (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩))
210209oveqd 7407 . . . . . . . . 9 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (𝑢(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑣) = (𝑢(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑣))
211 opex 5427 . . . . . . . . . . 11 ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ ∈ V
212211a1i 11 . . . . . . . . . 10 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ ∈ V)
213 simpl 482 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
214213fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (1st ‘𝑎) = (1st ‘𝑢))
215 simpr 484 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
216215fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (2nd ‘𝑏) = (2nd ‘𝑣))
217214, 216oveq12d 7408 . . . . . . . . . . . . 13 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑢) · (2nd ‘𝑣)))
218215fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (1st ‘𝑏) = (1st ‘𝑣))
219213fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (2nd ‘𝑎) = (2nd ‘𝑢))
220218, 219oveq12d 7408 . . . . . . . . . . . . 13 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑣) · (2nd ‘𝑢)))
221217, 220oveq12d 7408 . . . . . . . . . . . 12 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))))
222219, 216oveq12d 7408 . . . . . . . . . . . 12 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((2nd ‘𝑎) · (2nd ‘𝑏)) = ((2nd ‘𝑢) · (2nd ‘𝑣)))
223221, 222opeq12d 4848 . . . . . . . . . . 11 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩ = ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩)
224223, 21ovmpoga 7546 . . . . . . . . . 10 ((𝑢 ∈ (𝐵 × 𝑆) ∧ 𝑣 ∈ (𝐵 × 𝑆) ∧ ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ ∈ V) → (𝑢(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑣) = ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩)
22567, 72, 212, 224syl3anc 1373 . . . . . . . . 9 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (𝑢(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑣) = ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩)
226210, 225eqtrd 2765 . . . . . . . 8 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (𝑢(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑣) = ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩)
227209oveqd 7407 . . . . . . . . 9 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞) = (𝑝(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑞))
228 opex 5427 . . . . . . . . . . 11 ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩ ∈ V
229228a1i 11 . . . . . . . . . 10 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩ ∈ V)
230 simpl 482 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → 𝑎 = 𝑝)
231230fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (1st ‘𝑎) = (1st ‘𝑝))
232 simpr 484 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → 𝑏 = 𝑞)
233232fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (2nd ‘𝑏) = (2nd ‘𝑞))
234231, 233oveq12d 7408 . . . . . . . . . . . . 13 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑝) · (2nd ‘𝑞)))
235232fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (1st ‘𝑏) = (1st ‘𝑞))
236230fveq2d 6865 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (2nd ‘𝑎) = (2nd ‘𝑝))
237235, 236oveq12d 7408 . . . . . . . . . . . . 13 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑞) · (2nd ‘𝑝)))
238234, 237oveq12d 7408 . . . . . . . . . . . 12 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))))
239236, 233oveq12d 7408 . . . . . . . . . . . 12 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → ((2nd ‘𝑎) · (2nd ‘𝑏)) = ((2nd ‘𝑝) · (2nd ‘𝑞)))
240238, 239opeq12d 4848 . . . . . . . . . . 11 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩ = ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
241240, 21ovmpoga 7546 . . . . . . . . . 10 ((𝑝 ∈ (𝐵 × 𝑆) ∧ 𝑞 ∈ (𝐵 × 𝑆) ∧ ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩ ∈ V) → (𝑝(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑞) = ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
24285, 89, 229, 241syl3anc 1373 . . . . . . . . 9 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (𝑝(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑞) = ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
243227, 242eqtrd 2765 . . . . . . . 8 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞) = ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
244226, 243breq12d 5123 . . . . . . 7 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → ((𝑢(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑣) ∼ (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞) ↔ ⟨(((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))), ((2nd ‘𝑢) · (2nd ‘𝑣))⟩ ∼ ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩))
245201, 244mpbird 257 . . . . . 6 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → (𝑢(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑣) ∼ (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞))
246245anasss 466 . . . . 5 ((𝜑 ∧ (𝑢 ∼ 𝑝 ∧ 𝑣 ∼ 𝑞)) → (𝑢(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑣) ∼ (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞))
247246ex 412 . . . 4 (𝜑 → ((𝑢 ∼ 𝑝 ∧ 𝑣 ∼ 𝑞) → (𝑢(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑣) ∼ (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞)))
248208oveqd 7407 . . . . . . 7 (𝜑 → (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞) = (𝑝(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑞))
249248ad2antrr 726 . . . . . 6 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞) = (𝑝(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑞))
250 simplr 768 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑝 ∈ (𝐵 × 𝑆))
251 simpr 484 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑞 ∈ (𝐵 × 𝑆))
252228a1i 11 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩ ∈ V)
253250, 251, 252, 241syl3anc 1373 . . . . . . 7 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (𝑝(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑞) = ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩)
25462ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑅 ∈ Grp)
25564ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑅 ∈ Ring)
256250, 87syl 17 . . . . . . . . . 10 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (1st ‘𝑝) ∈ 𝐵)
25731ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑆 ⊆ 𝐵)
258251, 91syl 17 . . . . . . . . . . 11 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (2nd ‘𝑞) ∈ 𝑆)
259257, 258sseldd 3950 . . . . . . . . . 10 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (2nd ‘𝑞) ∈ 𝐵)
2608, 10, 255, 256, 259ringcld 20176 . . . . . . . . 9 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ((1st ‘𝑝) · (2nd ‘𝑞)) ∈ 𝐵)
261251, 95syl 17 . . . . . . . . . 10 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (1st ‘𝑞) ∈ 𝐵)
262250, 97syl 17 . . . . . . . . . . 11 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (2nd ‘𝑝) ∈ 𝑆)
263257, 262sseldd 3950 . . . . . . . . . 10 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (2nd ‘𝑝) ∈ 𝐵)
2648, 10, 255, 261, 263ringcld 20176 . . . . . . . . 9 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ((1st ‘𝑞) · (2nd ‘𝑝)) ∈ 𝐵)
2658, 12, 254, 260, 264grpcld 18886 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) ∈ 𝐵)
26627ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
267266, 262, 258, 106syl3anc 1373 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ((2nd ‘𝑝) · (2nd ‘𝑞)) ∈ 𝑆)
268265, 267opelxpd 5680 . . . . . . 7 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩ ∈ (𝐵 × 𝑆))
269253, 268eqeltrd 2829 . . . . . 6 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (𝑝(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)𝑞) ∈ (𝐵 × 𝑆))
270249, 269eqeltrd 2829 . . . . 5 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞) ∈ (𝐵 × 𝑆))
271270anasss 466 . . . 4 ((𝜑 ∧ (𝑝 ∈ (𝐵 × 𝑆) ∧ 𝑞 ∈ (𝐵 × 𝑆))) → (𝑝(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))𝑞) ∈ (𝐵 × 𝑆))
272 rlocaddval.10 . . . 4 ⊕ = (+g‘𝐿)
27333, 48, 50, 56, 247, 271, 207, 272qusaddval 17523 . . 3 ((𝜑 ∧ ⟨𝐸, 𝐺⟩ ∈ (𝐵 × 𝑆) ∧ ⟨𝐹, 𝐻⟩ ∈ (𝐵 × 𝑆)) → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [(⟨𝐸, 𝐺⟩(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))⟨𝐹, 𝐻⟩)] ∼ )
2743, 6, 273mpd3an23 1465 . 2 (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [(⟨𝐸, 𝐺⟩(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))⟨𝐹, 𝐻⟩)] ∼ )
275208oveqd 7407 . . . 4 (𝜑 → (⟨𝐸, 𝐺⟩(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))⟨𝐹, 𝐻⟩) = (⟨𝐸, 𝐺⟩(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟨𝐹, 𝐻⟩))
27621a1i 11 . . . . 5 (𝜑 → (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) = (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩))
277 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝑎 = ⟨𝐸, 𝐺⟩)
278277fveq2d 6865 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘𝑎) = (1st ‘⟨𝐸, 𝐺⟩))
2791adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐸 ∈ 𝐵)
2802adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐺 ∈ 𝑆)
281 op1stg 7983 . . . . . . . . . 10 ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (1st ‘⟨𝐸, 𝐺⟩) = 𝐸)
282279, 280, 281syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘⟨𝐸, 𝐺⟩) = 𝐸)
283278, 282eqtrd 2765 . . . . . . . 8 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘𝑎) = 𝐸)
284 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝑏 = ⟨𝐹, 𝐻⟩)
285284fveq2d 6865 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐻⟩))
2864adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐹 ∈ 𝐵)
2875adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐻 ∈ 𝑆)
288 op2ndg 7984 . . . . . . . . . 10 ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (2nd ‘⟨𝐹, 𝐻⟩) = 𝐻)
289286, 287, 288syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘⟨𝐹, 𝐻⟩) = 𝐻)
290285, 289eqtrd 2765 . . . . . . . 8 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘𝑏) = 𝐻)
291283, 290oveq12d 7408 . . . . . . 7 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
292284fveq2d 6865 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐻⟩))
293 op1stg 7983 . . . . . . . . . 10 ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (1st ‘⟨𝐹, 𝐻⟩) = 𝐹)
294286, 287, 293syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘⟨𝐹, 𝐻⟩) = 𝐹)
295292, 294eqtrd 2765 . . . . . . . 8 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘𝑏) = 𝐹)
296277fveq2d 6865 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘𝑎) = (2nd ‘⟨𝐸, 𝐺⟩))
297 op2ndg 7984 . . . . . . . . . 10 ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (2nd ‘⟨𝐸, 𝐺⟩) = 𝐺)
298279, 280, 297syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘⟨𝐸, 𝐺⟩) = 𝐺)
299296, 298eqtrd 2765 . . . . . . . 8 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘𝑎) = 𝐺)
300295, 299oveq12d 7408 . . . . . . 7 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐹 · 𝐺))
301291, 300oveq12d 7408 . . . . . 6 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))) = ((𝐸 · 𝐻) + (𝐹 · 𝐺)))
302299, 290oveq12d 7408 . . . . . 6 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → ((2nd ‘𝑎) · (2nd ‘𝑏)) = (𝐺 · 𝐻))
303301, 302opeq12d 4848 . . . . 5 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩ = ⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩)
304 opex 5427 . . . . . 6 ⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩ ∈ V
305304a1i 11 . . . . 5 (𝜑 → ⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩ ∈ V)
306276, 303, 3, 6, 305ovmpod 7544 . . . 4 (𝜑 → (⟨𝐸, 𝐺⟩(𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟨𝐹, 𝐻⟩) = ⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩)
307275, 306eqtrd 2765 . . 3 (𝜑 → (⟨𝐸, 𝐺⟩(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))⟨𝐹, 𝐻⟩) = ⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩)
308307eceq1d 8714 . 2 (𝜑 → [(⟨𝐸, 𝐺⟩(+g‘(({⟨(Base‘ndx), (𝐵 × 𝑆)⟩, ⟨(+g‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑅)), 𝑎 ∈ (𝐵 × 𝑆) ↦ ⟨(𝑘( ·𝑠 ‘𝑅)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑅) ×t ((TopSet‘𝑅) ↾t 𝑆))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ((1st ‘𝑎) · (2nd ‘𝑏))(le‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ (((1st ‘𝑎) · (2nd ‘𝑏))(dist‘𝑅)((1st ‘𝑏) · (2nd ‘𝑎))))⟩}))⟨𝐹, 𝐻⟩)] ∼ = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] ∼ )
309274, 308eqtrd 2765 1 (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] ∼ )
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  {csn 4592  {ctp 4596  ⟨cop 4598   class class class wbr 5110  {copab 5172   × cxp 5639  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1st c1st 7969  2nd c2nd 7970  [cec 8672  1c1 11076  2c2 12248  cdc 12656  ndxcnx 17170  Basecbs 17186  +gcplusg 17227  .rcmulr 17228  Scalarcsca 17230   ·𝑠 cvsca 17231  ·𝑖cip 17232  TopSetcts 17233  lecple 17234  distcds 17236   ↾t crest 17390  0gc0g 17409   /s cqus 17475  SubMndcsubmnd 18716  Grpcgrp 18872  -gcsg 18874  CMndccmn 19717  Abelcabl 19718  mulGrpcmgp 20056  1rcur 20097  Ringcrg 20149  CRingccrg 20150   ×t ctx 23454   ~RL cerl 33211   RLocal crloc 33212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-ec 8676  df-qs 8680  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-0g 17411  df-imas 17478  df-qus 17479  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-cring 20152  df-erl 33213  df-rloc 33214
This theorem is referenced by:  rloccring  33228  rloc0g  33229  rlocf1  33231  zringfrac  33532
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