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Theorem rlocaddval 33058
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 5717 . . 3 (𝜑 → ⟨𝐸, 𝐺⟩ ∈ (𝐵 × 𝑆))
4 rlocaddval.7 . . . 4 (𝜑 → 𝐹 ∈ 𝐵)
5 rlocaddval.9 . . . 4 (𝜑 → 𝐻 ∈ 𝑆)
64, 5opelxpd 5717 . . 3 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝐵 × 𝑆))
7 rlocaddval.4 . . . . 5 𝐿 = (𝑅 RLocal 𝑆)
8 rlocaddval.1 . . . . . 6 𝐵 = (Base‘𝑅)
9 eqid 2725 . . . . . 6 (0g‘𝑅) = (0g‘𝑅)
10 rlocaddval.2 . . . . . 6 · = (.r‘𝑅)
11 eqid 2725 . . . . . 6 (-g‘𝑅) = (-g‘𝑅)
12 rlocaddval.3 . . . . . 6 + = (+g‘𝑅)
13 eqid 2725 . . . . . 6 (le‘𝑅) = (le‘𝑅)
14 eqid 2725 . . . . . 6 (Scalar‘𝑅) = (Scalar‘𝑅)
15 eqid 2725 . . . . . 6 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
16 eqid 2725 . . . . . 6 ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
17 eqid 2725 . . . . . 6 (𝐵 × 𝑆) = (𝐵 × 𝑆)
18 rlocaddval.5 . . . . . 6 ∼ = (𝑅 ~RL 𝑆)
19 eqid 2725 . . . . . 6 (TopSet‘𝑅) = (TopSet‘𝑅)
20 eqid 2725 . . . . . 6 (dist‘𝑅) = (dist‘𝑅)
21 eqid 2725 . . . . . 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 2725 . . . . . 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 2725 . . . . . 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 2725 . . . . . 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 2725 . . . . . 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 2725 . . . . . . . . 9 (mulGrp‘𝑅) = (mulGrp‘𝑅)
2928, 8mgpbas 20092 . . . . . . . 8 𝐵 = (Base‘(mulGrp‘𝑅))
3029submss 18769 . . . . . . 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 33049 . . . . 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 2726 . . . . . 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 2725 . . . . . . 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 17436 . . . . . 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 17186 . . . . . 6 Base = Slot (Base‘ndx)
38 snsstp1 4821 . . . . . . 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 4170 . . . . . . . 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 4170 . . . . . . . 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 3986 . . . . . . 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 3986 . . . . . 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 6910 . . . . . . . 8 𝐵 ∈ V
4443a1i 11 . . . . . . 7 (𝜑 → 𝐵 ∈ V)
4544, 27xpexd 7754 . . . . . 6 (𝜑 → (𝐵 × 𝑆) ∈ V)
46 eqid 2725 . . . . . 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 17177 . . . . 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 2731 . . . 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 2725 . . . . 5 (1r‘𝑅) = (1r‘𝑅)
508, 9, 49, 10, 11, 17, 18, 26, 27erler 33055 . . . 4 (𝜑 → ∼ Er (𝐵 × 𝑆))
51 tpex 7750 . . . . . . 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 7750 . . . . . . 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 7749 . . . . . 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 7750 . . . . . 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 7749 . . . . 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 724 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑆 ⊆ 𝐵)
5857ad2antrr 724 . . . . . . . . . . 11 (((((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) ∧ 𝑓 ∈ 𝑆) ∧ (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅)) → 𝑆 ⊆ 𝐵)
5958ad2antrr 724 . . . . . . . . . 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 2726 . . . . . . . . . 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 2726 . . . . . . . . . 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 20199 . . . . . . . . . . . 12 (𝜑 → 𝑅 ∈ Grp)
6362ad6antr 734 . . . . . . . . . . 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 20198 . . . . . . . . . . . . 13 (𝜑 → 𝑅 ∈ Ring)
6564ad6antr 734 . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑢 ∼ 𝑝)
678, 18, 57, 66erlcl1 33050 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑢 ∈ (𝐵 × 𝑆))
6867ad4antr 730 . . . . . . . . . . . . 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 8026 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑣 ∼ 𝑞)
728, 18, 57, 71erlcl1 33050 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑣 ∈ (𝐵 × 𝑆))
7372ad4antr 730 . . . . . . . . . . . . . 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 8027 . . . . . . . . . . . . . 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 3977 . . . . . . . . . . . 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 20211 . . . . . . . . . . 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 8026 . . . . . . . . . . . . 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 8027 . . . . . . . . . . . . . 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 3977 . . . . . . . . . . . 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 20211 . . . . . . . . . . 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 18912 . . . . . . . . . 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 33051 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑝 ∈ (𝐵 × 𝑆))
8685ad4antr 730 . . . . . . . . . . . . 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 8026 . . . . . . . . . . . . 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 33051 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → 𝑞 ∈ (𝐵 × 𝑆))
9089ad4antr 730 . . . . . . . . . . . . . 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 8027 . . . . . . . . . . . . . 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 3977 . . . . . . . . . . . 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 20211 . . . . . . . . . . 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 8026 . . . . . . . . . . . . 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 8027 . . . . . . . . . . . . . 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 3977 . . . . . . . . . . . 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 20211 . . . . . . . . . . 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 18912 . . . . . . . . . 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 734 . . . . . . . . . . 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 20090 . . . . . . . . . . . 12 · = (+g‘(mulGrp‘𝑅))
104103submcl 18772 . . . . . . . . . . 11 ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ (2nd ‘𝑢) ∈ 𝑆 ∧ (2nd ‘𝑣) ∈ 𝑆) → ((2nd ‘𝑢) · (2nd ‘𝑣)) ∈ 𝑆)
105102, 81, 75, 104syl3anc 1368 . . . . . . . . . 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 18772 . . . . . . . . . . 11 ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ (2nd ‘𝑝) ∈ 𝑆 ∧ (2nd ‘𝑞) ∈ 𝑆) → ((2nd ‘𝑝) · (2nd ‘𝑞)) ∈ 𝑆)
107102, 98, 92, 106syl3anc 1368 . . . . . . . . . 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 782 . . . . . . . . . . 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 767 . . . . . . . . . . 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 18772 . . . . . . . . . . 11 ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ 𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) → (𝑓 · 𝑔) ∈ 𝑆)
111102, 108, 109, 110syl3anc 1368 . . . . . . . . . 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 3977 . . . . . . . . . . . . . 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 20213 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 3977 . . . . . . . . . . . . . 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 20213 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 7437 . . . . . . . . . . . 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 7435 . . . . . . . . . . 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 3977 . . . . . . . . . . . . 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 3977 . . . . . . . . . . . . 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 20211 . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 18912 . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 18912 . . . . . . . . . . . 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 20255 . . . . . . . . . . 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 20212 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 20212 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 7437 . . . . . . . . . . . 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 20231 . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . 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 19777 . . . . . . . . . . . . 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 1376 . . . . . . . . . . . 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 20193 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
14326, 142syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
144143ad6antr 734 . . . . . . . . . . . . . . . . 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 32842 . . . . . . . . . . . . . . . 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 32842 . . . . . . . . . . . . . . . . 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 19765 . . . . . . . . . . . . . . . . . . . 20 (((mulGrp‘𝑅) ∈ CMnd ∧ (2nd ‘𝑣) ∈ 𝐵 ∧ (2nd ‘𝑞) ∈ 𝐵) → ((2nd ‘𝑣) · (2nd ‘𝑞)) = ((2nd ‘𝑞) · (2nd ‘𝑣)))
148144, 76, 93, 147syl3anc 1368 . . . . . . . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . . . 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 20255 . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . 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 20255 . . . . . . . . . . . . . . . 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 20244 . . . . . . . . . . . . . . . 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 32843 . . . . . . . . . . . . . . . 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 32843 . . . . . . . . . . . . . . . . 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 19765 . . . . . . . . . . . . . . . . . . . 20 (((mulGrp‘𝑅) ∈ CMnd ∧ (2nd ‘𝑝) ∈ 𝐵 ∧ (2nd ‘𝑢) ∈ 𝐵) → ((2nd ‘𝑝) · (2nd ‘𝑢)) = ((2nd ‘𝑢) · (2nd ‘𝑝)))
170144, 99, 82, 169syl3anc 1368 . . . . . . . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . . . 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 20255 . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . 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 20211 . . . . . . . . . . . . . . . . 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 20255 . . . . . . . . . . . . . . . 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 20244 . . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . 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 18930 . . . . . . . . . . . . . . 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 18934 . . . . . . . . . . . . 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 33053 . . . . . . . . 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 724 . . . . . . . . . 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 33052 . . . . . . . . 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 3151 . . . . . . . 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 33052 . . . . . . . 8 (((𝜑 ∧ 𝑢 ∼ 𝑝) ∧ 𝑣 ∼ 𝑞) → ∃𝑓 ∈ 𝑆 (𝑓 · (((1st ‘𝑢) · (2nd ‘𝑝))(-g‘𝑅)((1st ‘𝑝) · (2nd ‘𝑢)))) = (0g‘𝑅))
201199, 200r19.29a 3151 . . . . . . 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 17263 . . . . . . . . . . . 12 +g = Slot (+g‘ndx)
203 snsstp2 4822 . . . . . . . . . . . . 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 3986 . . . . . . . . . . . 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 8081 . . . . . . . . . . . . 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 582 . . . . . . . . . . . 12 (𝜑 → (𝑎 ∈ (𝐵 × 𝑆), 𝑏 ∈ (𝐵 × 𝑆) ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩) ∈ V)
207 eqid 2725 . . . . . . . . . . . 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 17177 . . . . . . . . . . 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 724 . . . . . . . . . 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 7436 . . . . . . . . 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 5466 . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
214213fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (1st ‘𝑎) = (1st ‘𝑢))
215 simpr 483 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
216215fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (2nd ‘𝑏) = (2nd ‘𝑣))
217214, 216oveq12d 7437 . . . . . . . . . . . . 13 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑢) · (2nd ‘𝑣)))
218215fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (1st ‘𝑏) = (1st ‘𝑣))
219213fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (2nd ‘𝑎) = (2nd ‘𝑢))
220218, 219oveq12d 7437 . . . . . . . . . . . . 13 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑣) · (2nd ‘𝑢)))
221217, 220oveq12d 7437 . . . . . . . . . . . 12 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑢) · (2nd ‘𝑣)) + ((1st ‘𝑣) · (2nd ‘𝑢))))
222219, 216oveq12d 7437 . . . . . . . . . . . 12 ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((2nd ‘𝑎) · (2nd ‘𝑏)) = ((2nd ‘𝑢) · (2nd ‘𝑣)))
223221, 222opeq12d 4883 . . . . . . . . . . 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 7575 . . . . . . . . . 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 1368 . . . . . . . . 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 7436 . . . . . . . . 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 5466 . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → 𝑎 = 𝑝)
231230fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (1st ‘𝑎) = (1st ‘𝑝))
232 simpr 483 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → 𝑏 = 𝑞)
233232fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (2nd ‘𝑏) = (2nd ‘𝑞))
234231, 233oveq12d 7437 . . . . . . . . . . . . 13 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑝) · (2nd ‘𝑞)))
235232fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (1st ‘𝑏) = (1st ‘𝑞))
236230fveq2d 6900 . . . . . . . . . . . . . 14 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (2nd ‘𝑎) = (2nd ‘𝑝))
237235, 236oveq12d 7437 . . . . . . . . . . . . 13 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑞) · (2nd ‘𝑝)))
238234, 237oveq12d 7437 . . . . . . . . . . . 12 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → (((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))))
239236, 233oveq12d 7437 . . . . . . . . . . . 12 ((𝑎 = 𝑝 ∧ 𝑏 = 𝑞) → ((2nd ‘𝑎) · (2nd ‘𝑏)) = ((2nd ‘𝑝) · (2nd ‘𝑞)))
240238, 239opeq12d 4883 . . . . . . . . . . 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 7575 . . . . . . . . . 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 1368 . . . . . . . . 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 5162 . . . . . . 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 256 . . . . . 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 465 . . . . 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 411 . . . 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 7436 . . . . . . 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 724 . . . . . 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 767 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑝 ∈ (𝐵 × 𝑆))
251 simpr 483 . . . . . . . 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 1368 . . . . . . 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 724 . . . . . . . . 9 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑅 ∈ Grp)
25564ad2antrr 724 . . . . . . . . . 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 724 . . . . . . . . . . 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 3977 . . . . . . . . . 10 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (2nd ‘𝑞) ∈ 𝐵)
2608, 10, 255, 256, 259ringcld 20211 . . . . . . . . 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 3977 . . . . . . . . . 10 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (2nd ‘𝑝) ∈ 𝐵)
2648, 10, 255, 261, 263ringcld 20211 . . . . . . . . 9 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ((1st ‘𝑞) · (2nd ‘𝑝)) ∈ 𝐵)
2658, 12, 254, 260, 264grpcld 18912 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → (((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))) ∈ 𝐵)
26627ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
267266, 262, 258, 106syl3anc 1368 . . . . . . . 8 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ((2nd ‘𝑝) · (2nd ‘𝑞)) ∈ 𝑆)
268265, 267opelxpd 5717 . . . . . . 7 (((𝜑 ∧ 𝑝 ∈ (𝐵 × 𝑆)) ∧ 𝑞 ∈ (𝐵 × 𝑆)) → ⟨(((1st ‘𝑝) · (2nd ‘𝑞)) + ((1st ‘𝑞) · (2nd ‘𝑝))), ((2nd ‘𝑝) · (2nd ‘𝑞))⟩ ∈ (𝐵 × 𝑆))
269253, 268eqeltrd 2825 . . . . . 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 2825 . . . . 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 465 . . . 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 17538 . . 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 1459 . 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 7436 . . . 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 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝑎 = ⟨𝐸, 𝐺⟩)
278277fveq2d 6900 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘𝑎) = (1st ‘⟨𝐸, 𝐺⟩))
2791adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐸 ∈ 𝐵)
2802adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐺 ∈ 𝑆)
281 op1stg 8006 . . . . . . . . . 10 ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (1st ‘⟨𝐸, 𝐺⟩) = 𝐸)
282279, 280, 281syl2anc 582 . . . . . . . . 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 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝑏 = ⟨𝐹, 𝐻⟩)
285284fveq2d 6900 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝐻⟩))
2864adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐹 ∈ 𝐵)
2875adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → 𝐻 ∈ 𝑆)
288 op2ndg 8007 . . . . . . . . . 10 ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (2nd ‘⟨𝐹, 𝐻⟩) = 𝐻)
289286, 287, 288syl2anc 582 . . . . . . . . 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 7437 . . . . . . 7 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
292284fveq2d 6900 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝐻⟩))
293 op1stg 8006 . . . . . . . . . 10 ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (1st ‘⟨𝐹, 𝐻⟩) = 𝐹)
294286, 287, 293syl2anc 582 . . . . . . . . 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 6900 . . . . . . . . 9 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (2nd ‘𝑎) = (2nd ‘⟨𝐸, 𝐺⟩))
297 op2ndg 8007 . . . . . . . . . 10 ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (2nd ‘⟨𝐸, 𝐺⟩) = 𝐺)
298279, 280, 297syl2anc 582 . . . . . . . . 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 7437 . . . . . . 7 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐹 · 𝐺))
301291, 300oveq12d 7437 . . . . . 6 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → (((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))) = ((𝐸 · 𝐻) + (𝐹 · 𝐺)))
302299, 290oveq12d 7437 . . . . . 6 ((𝜑 ∧ (𝑎 = ⟨𝐸, 𝐺⟩ ∧ 𝑏 = ⟨𝐹, 𝐻⟩)) → ((2nd ‘𝑎) · (2nd ‘𝑏)) = (𝐺 · 𝐻))
303301, 302opeq12d 4883 . . . . 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 5466 . . . . . 6 ⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩ ∈ V
305304a1i 11 . . . . 5 (𝜑 → ⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩ ∈ V)
306276, 303, 3, 6, 305ovmpod 7573 . . . 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 8764 . 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  {csn 4630  {ctp 4634  ⟨cop 4636   class class class wbr 5149  {copab 5211   × cxp 5676  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  1st c1st 7992  2nd c2nd 7993  [cec 8723  1c1 11141  2c2 12300  cdc 12710  ndxcnx 17165  Basecbs 17183  +gcplusg 17236  .rcmulr 17237  Scalarcsca 17239   ·𝑠 cvsca 17240  ·𝑖cip 17241  TopSetcts 17242  lecple 17243  distcds 17245   ↾t crest 17405  0gc0g 17424   /s cqus 17490  SubMndcsubmnd 18742  Grpcgrp 18898  -gcsg 18900  CMndccmn 19747  Abelcabl 19748  mulGrpcmgp 20086  1rcur 20133  Ringcrg 20185  CRingccrg 20186   ×t ctx 23508   ~RL cerl 33043   RLocal crloc 33044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-ec 8727  df-qs 8731  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-0g 17426  df-imas 17493  df-qus 17494  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-grp 18901  df-minusg 18902  df-sbg 18903  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-erl 33045  df-rloc 33046
This theorem is referenced by:  rloccring  33060  rloc0g  33061  rlocf1  33063  zringfrac  33369
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