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Theorem prdsval 16047
Description: Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
prdsval.p 𝑃 = (𝑆Xs𝑅)
prdsval.k 𝐾 = (Base‘𝑆)
prdsval.i (𝜑 → dom 𝑅 = 𝐼)
prdsval.b (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
prdsval.a (𝜑+ = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
prdsval.t (𝜑× = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
prdsval.m (𝜑· = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
prdsval.j (𝜑, = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
prdsval.o (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
prdsval.l (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
prdsval.d (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
prdsval.h (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
prdsval.x (𝜑 = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
prdsval.s (𝜑𝑆𝑊)
prdsval.r (𝜑𝑅𝑍)
Assertion
Ref Expression
prdsval (𝜑𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))
Distinct variable groups:   𝑎,𝑐,𝑑,𝑒,𝑓,𝑔,𝐵   𝐻,𝑎,𝑐,𝑑,𝑒   𝑥,𝑎,𝜑,𝑐,𝑑,𝑒,𝑓,𝑔   𝑥,𝐼   𝑅,𝑎,𝑐,𝑑,𝑒,𝑓,𝑔,𝑥   𝑆,𝑎,𝑐,𝑑,𝑒,𝑓,𝑔,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑃(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   + (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   · (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   × (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝐻(𝑥,𝑓,𝑔)   , (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝐼(𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝐾(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   (𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑂(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑊(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)   𝑍(𝑥,𝑒,𝑓,𝑔,𝑎,𝑐,𝑑)

Proof of Theorem prdsval
Dummy variables 𝑟 𝑠 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsval.p . 2 𝑃 = (𝑆Xs𝑅)
2 df-prds 16040 . . . 4 Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
32a1i 11 . . 3 (𝜑Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))))
4 vex 3192 . . . . . . . . . . . 12 𝑟 ∈ V
54rnex 7054 . . . . . . . . . . 11 ran 𝑟 ∈ V
65uniex 6913 . . . . . . . . . 10 ran 𝑟 ∈ V
76rnex 7054 . . . . . . . . 9 ran ran 𝑟 ∈ V
87uniex 6913 . . . . . . . 8 ran ran 𝑟 ∈ V
9 baseid 15851 . . . . . . . . . . . 12 Base = Slot (Base‘ndx)
109strfvss 15813 . . . . . . . . . . 11 (Base‘(𝑟𝑥)) ⊆ ran (𝑟𝑥)
11 fvssunirn 6179 . . . . . . . . . . . 12 (𝑟𝑥) ⊆ ran 𝑟
12 rnss 5319 . . . . . . . . . . . 12 ((𝑟𝑥) ⊆ ran 𝑟 → ran (𝑟𝑥) ⊆ ran ran 𝑟)
13 uniss 4429 . . . . . . . . . . . 12 (ran (𝑟𝑥) ⊆ ran ran 𝑟 ran (𝑟𝑥) ⊆ ran ran 𝑟)
1411, 12, 13mp2b 10 . . . . . . . . . . 11 ran (𝑟𝑥) ⊆ ran ran 𝑟
1510, 14sstri 3596 . . . . . . . . . 10 (Base‘(𝑟𝑥)) ⊆ ran ran 𝑟
1615rgenw 2919 . . . . . . . . 9 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ran ran 𝑟
17 iunss 4532 . . . . . . . . 9 ( 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ran ran 𝑟 ↔ ∀𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ran ran 𝑟)
1816, 17mpbir 221 . . . . . . . 8 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ran ran 𝑟
198, 18ssexi 4768 . . . . . . 7 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∈ V
20 ixpssmap2g 7889 . . . . . . 7 ( 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∈ V → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ( 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ↑𝑚 dom 𝑟))
2119, 20ax-mp 5 . . . . . 6 X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ( 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ↑𝑚 dom 𝑟)
22 ovex 6638 . . . . . . 7 ( 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ↑𝑚 dom 𝑟) ∈ V
2322ssex 4767 . . . . . 6 (X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ( 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ↑𝑚 dom 𝑟) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∈ V)
2421, 23mp1i 13 . . . . 5 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∈ V)
25 simpr 477 . . . . . . . . 9 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
2625fveq1d 6155 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑟𝑥) = (𝑅𝑥))
2726fveq2d 6157 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Base‘(𝑟𝑥)) = (Base‘(𝑅𝑥)))
2827ixpeq2dv 7876 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥𝐼 (Base‘(𝑟𝑥)) = X𝑥𝐼 (Base‘(𝑅𝑥)))
2925dmeqd 5291 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑅)
30 prdsval.i . . . . . . . . 9 (𝜑 → dom 𝑅 = 𝐼)
3130ad2antrr 761 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑅 = 𝐼)
3229, 31eqtrd 2655 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = 𝐼)
3332ixpeq1d 7872 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) = X𝑥𝐼 (Base‘(𝑟𝑥)))
34 prdsval.b . . . . . . 7 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
3534ad2antrr 761 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
3628, 33, 353eqtr4d 2665 . . . . 5 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) = 𝐵)
37 ovssunirn 6641 . . . . . . . . . . . . . . 15 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ran (Hom ‘(𝑟𝑥))
38 df-hom 15898 . . . . . . . . . . . . . . . . . 18 Hom = Slot 14
3938strfvss 15813 . . . . . . . . . . . . . . . . 17 (Hom ‘(𝑟𝑥)) ⊆ ran (𝑟𝑥)
4039, 14sstri 3596 . . . . . . . . . . . . . . . 16 (Hom ‘(𝑟𝑥)) ⊆ ran ran 𝑟
41 rnss 5319 . . . . . . . . . . . . . . . 16 ((Hom ‘(𝑟𝑥)) ⊆ ran ran 𝑟 → ran (Hom ‘(𝑟𝑥)) ⊆ ran ran ran 𝑟)
42 uniss 4429 . . . . . . . . . . . . . . . 16 (ran (Hom ‘(𝑟𝑥)) ⊆ ran ran ran 𝑟 ran (Hom ‘(𝑟𝑥)) ⊆ ran ran ran 𝑟)
4340, 41, 42mp2b 10 . . . . . . . . . . . . . . 15 ran (Hom ‘(𝑟𝑥)) ⊆ ran ran ran 𝑟
4437, 43sstri 3596 . . . . . . . . . . . . . 14 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑟
4544rgenw 2919 . . . . . . . . . . . . 13 𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑟
46 ss2ixp 7873 . . . . . . . . . . . . 13 (∀𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑟X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ X𝑥 ∈ dom 𝑟 ran ran ran 𝑟)
4745, 46ax-mp 5 . . . . . . . . . . . 12 X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ X𝑥 ∈ dom 𝑟 ran ran ran 𝑟
484dmex 7053 . . . . . . . . . . . . 13 dom 𝑟 ∈ V
498rnex 7054 . . . . . . . . . . . . . 14 ran ran ran 𝑟 ∈ V
5049uniex 6913 . . . . . . . . . . . . 13 ran ran ran 𝑟 ∈ V
5148, 50ixpconst 7870 . . . . . . . . . . . 12 X𝑥 ∈ dom 𝑟 ran ran ran 𝑟 = ( ran ran ran 𝑟𝑚 dom 𝑟)
5247, 51sseqtri 3621 . . . . . . . . . . 11 X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑟𝑚 dom 𝑟)
53 ovex 6638 . . . . . . . . . . . 12 ( ran ran ran 𝑟𝑚 dom 𝑟) ∈ V
5453elpw2 4793 . . . . . . . . . . 11 (X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟) ↔ X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑟𝑚 dom 𝑟))
5552, 54mpbir 221 . . . . . . . . . 10 X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟)
5655rgen2w 2920 . . . . . . . . 9 𝑓𝑣𝑔𝑣 X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟)
57 eqid 2621 . . . . . . . . . 10 (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) = (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)))
5857fmpt2 7189 . . . . . . . . 9 (∀𝑓𝑣𝑔𝑣 X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟) ↔ (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))):(𝑣 × 𝑣)⟶𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟))
5956, 58mpbi 220 . . . . . . . 8 (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))):(𝑣 × 𝑣)⟶𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟)
60 vex 3192 . . . . . . . . 9 𝑣 ∈ V
6160, 60xpex 6922 . . . . . . . 8 (𝑣 × 𝑣) ∈ V
6253pwex 4813 . . . . . . . 8 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟) ∈ V
63 fex2 7075 . . . . . . . 8 (((𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))):(𝑣 × 𝑣)⟶𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟) ∧ (𝑣 × 𝑣) ∈ V ∧ 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟) ∈ V) → (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) ∈ V)
6459, 61, 62, 63mp3an 1421 . . . . . . 7 (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) ∈ V
6564a1i 11 . . . . . 6 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) ∈ V)
66 simpr 477 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
6732adantr 481 . . . . . . . . . 10 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → dom 𝑟 = 𝐼)
6867ixpeq1d 7872 . . . . . . . . 9 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)))
6926fveq2d 6157 . . . . . . . . . . . 12 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Hom ‘(𝑟𝑥)) = (Hom ‘(𝑅𝑥)))
7069oveqd 6627 . . . . . . . . . . 11 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7170ixpeq2dv 7876 . . . . . . . . . 10 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7271adantr 481 . . . . . . . . 9 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7368, 72eqtrd 2655 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7466, 66, 73mpt2eq123dv 6677 . . . . . . 7 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
75 prdsval.h . . . . . . . 8 (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
7675ad3antrrr 765 . . . . . . 7 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
7774, 76eqtr4d 2658 . . . . . 6 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) = 𝐻)
78 simplr 791 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑣 = 𝐵)
7978opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝐵⟩)
8026fveq2d 6157 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (+g‘(𝑟𝑥)) = (+g‘(𝑅𝑥)))
8180oveqd 6627 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))
8232, 81mpteq12dv 4698 . . . . . . . . . . . . . 14 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))
8382adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))
8466, 66, 83mpt2eq123dv 6677 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
8584adantr 481 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
86 prdsval.a . . . . . . . . . . . 12 (𝜑+ = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
8786ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
8885, 87eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥)))) = + )
8988opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩ = ⟨(+g‘ndx), + ⟩)
9026fveq2d 6157 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (.r‘(𝑟𝑥)) = (.r‘(𝑅𝑥)))
9190oveqd 6627 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))
9232, 91mpteq12dv 4698 . . . . . . . . . . . . . 14 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))
9392adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))
9466, 66, 93mpt2eq123dv 6677 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
9594adantr 481 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
96 prdsval.t . . . . . . . . . . . 12 (𝜑× = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
9796ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → × = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
9895, 97eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥)))) = × )
9998opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩ = ⟨(.r‘ndx), × ⟩)
10079, 89, 99tpeq123d 4258 . . . . . . . 8 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → {⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩})
101 simp-4r 806 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑠 = 𝑆)
102101opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(Scalar‘ndx), 𝑠⟩ = ⟨(Scalar‘ndx), 𝑆⟩)
103 simpllr 798 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑠 = 𝑆)
104103fveq2d 6157 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = (Base‘𝑆))
105 prdsval.k . . . . . . . . . . . . . 14 𝐾 = (Base‘𝑆)
106104, 105syl6eqr 2673 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = 𝐾)
10726fveq2d 6157 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑟𝑥)) = ( ·𝑠 ‘(𝑅𝑥)))
108107oveqd 6627 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥)) = (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))
10932, 108mpteq12dv 4698 . . . . . . . . . . . . . 14 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))
110109adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))
111106, 66, 110mpt2eq123dv 6677 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
112111adantr 481 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
113 prdsval.m . . . . . . . . . . . 12 (𝜑· = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
114113ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → · = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
115112, 114eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥)))) = · )
116115opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
11726fveq2d 6157 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (·𝑖‘(𝑟𝑥)) = (·𝑖‘(𝑅𝑥)))
118117oveqd 6627 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))
11932, 118mpteq12dv 4698 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))
120119adantr 481 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))
121103, 120oveq12d 6628 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))) = (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))
12266, 66, 121mpt2eq123dv 6677 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
123122adantr 481 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
124 prdsval.j . . . . . . . . . . . 12 (𝜑, = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
125124ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → , = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
126123, 125eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))))) = , )
127126opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩ = ⟨(·𝑖‘ndx), , ⟩)
128102, 116, 127tpeq123d 4258 . . . . . . . 8 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩} = {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})
129100, 128uneq12d 3751 . . . . . . 7 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
130 simpllr 798 . . . . . . . . . . . . 13 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑟 = 𝑅)
131130coeq2d 5249 . . . . . . . . . . . 12 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (TopOpen ∘ 𝑟) = (TopOpen ∘ 𝑅))
132131fveq2d 6157 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (∏t‘(TopOpen ∘ 𝑟)) = (∏t‘(TopOpen ∘ 𝑅)))
133 prdsval.o . . . . . . . . . . . 12 (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
134133ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
135132, 134eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (∏t‘(TopOpen ∘ 𝑟)) = 𝑂)
136135opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩ = ⟨(TopSet‘ndx), 𝑂⟩)
13766sseq2d 3617 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ({𝑓, 𝑔} ⊆ 𝑣 ↔ {𝑓, 𝑔} ⊆ 𝐵))
13826fveq2d 6157 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (le‘(𝑟𝑥)) = (le‘(𝑅𝑥)))
139138breqd 4629 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥) ↔ (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)))
14032, 139raleqbidv 3144 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥) ↔ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)))
141140adantr 481 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥) ↔ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)))
142137, 141anbi12d 746 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥)) ↔ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))))
143142opabbidv 4683 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
144143adantr 481 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
145 prdsval.l . . . . . . . . . . . 12 (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
146145ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
147144, 146eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))} = )
148147opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩ = ⟨(le‘ndx), ⟩)
14926fveq2d 6157 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (dist‘(𝑟𝑥)) = (dist‘(𝑅𝑥)))
150149oveqd 6627 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥)))
15132, 150mpteq12dv 4698 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))))
152151adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))))
153152rneqd 5318 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) = ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))))
154153uneq1d 3749 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}) = (ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}))
155154supeq1d 8304 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ) = sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))
15666, 66, 155mpt2eq123dv 6677 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
157156adantr 481 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
158 prdsval.d . . . . . . . . . . . 12 (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
159158ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
160157, 159eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = 𝐷)
161160opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩ = ⟨(dist‘ndx), 𝐷⟩)
162136, 148, 161tpeq123d 4258 . . . . . . . 8 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → {⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} = {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})
163 simpr 477 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → = 𝐻)
164163opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
16578sqxpeqd 5106 . . . . . . . . . . . 12 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑣 × 𝑣) = (𝐵 × 𝐵))
166163oveqd 6627 . . . . . . . . . . . . 13 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑐(2nd𝑎)) = (𝑐𝐻(2nd𝑎)))
167163fveq1d 6155 . . . . . . . . . . . . 13 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑎) = (𝐻𝑎))
16826fveq2d 6157 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (comp‘(𝑟𝑥)) = (comp‘(𝑅𝑥)))
169168oveqd 6627 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥)) = (⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥)))
170169oveqd 6627 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)) = ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))
17132, 170mpteq12dv 4698 . . . . . . . . . . . . . 14 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥))) = (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))
172171ad2antrr 761 . . . . . . . . . . . . 13 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥))) = (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))
173166, 167, 172mpt2eq123dv 6677 . . . . . . . . . . . 12 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))) = (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))
174165, 78, 173mpt2eq123dv 6677 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
175 prdsval.x . . . . . . . . . . . 12 (𝜑 = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
176175ad4antr 767 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
177174, 176eqtr4d 2658 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥))))) = )
178177opeq2d 4382 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩ = ⟨(comp‘ndx), ⟩)
179164, 178preq12d 4251 . . . . . . . 8 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩} = {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})
180162, 179uneq12d 3751 . . . . . . 7 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}) = ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩}))
181129, 180uneq12d 3751 . . . . . 6 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))
18265, 77, 181csbied2 3546 . . . . 5 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))
18324, 36, 182csbied2 3546 . . . 4 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))
184183anasss 678 . . 3 ((𝜑 ∧ (𝑠 = 𝑆𝑟 = 𝑅)) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))
185 prdsval.s . . . 4 (𝜑𝑆𝑊)
186 elex 3201 . . . 4 (𝑆𝑊𝑆 ∈ V)
187185, 186syl 17 . . 3 (𝜑𝑆 ∈ V)
188 prdsval.r . . . 4 (𝜑𝑅𝑍)
189 elex 3201 . . . 4 (𝑅𝑍𝑅 ∈ V)
190188, 189syl 17 . . 3 (𝜑𝑅 ∈ V)
191 tpex 6917 . . . . . 6 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
192 tpex 6917 . . . . . 6 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
193191, 192unex 6916 . . . . 5 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V
194 tpex 6917 . . . . . 6 {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∈ V
195 prex 4875 . . . . . 6 {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩} ∈ V
196194, 195unex 6916 . . . . 5 ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩}) ∈ V
197193, 196unex 6916 . . . 4 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})) ∈ V
198197a1i 11 . . 3 (𝜑 → (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})) ∈ V)
1993, 184, 187, 190, 198ovmpt2d 6748 . 2 (𝜑 → (𝑆Xs𝑅) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))
2001, 199syl5eq 2667 1 (𝜑𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  csb 3518  cun 3557  wss 3559  𝒫 cpw 4135  {csn 4153  {cpr 4155  {ctp 4157  cop 4159   cuni 4407   ciun 4490   class class class wbr 4618  {copab 4677  cmpt 4678   × cxp 5077  dom cdm 5079  ran crn 5080  ccom 5083  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  1st c1st 7118  2nd c2nd 7119  𝑚 cmap 7809  Xcixp 7860  supcsup 8298  0cc0 9888  1c1 9889  *cxr 10025   < clt 10026  4c4 11024  cdc 11445  ndxcnx 15789  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  Scalarcsca 15876   ·𝑠 cvsca 15877  ·𝑖cip 15878  TopSetcts 15879  lecple 15880  distcds 15882  Hom chom 15884  compcco 15885  TopOpenctopn 16014  tcpt 16031   Σg cgsu 16033  Xscprds 16038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-i2m1 9956  ax-1ne0 9957  ax-rrecex 9960  ax-cnre 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-map 7811  df-ixp 7861  df-sup 8300  df-nn 10973  df-ndx 15795  df-slot 15796  df-base 15797  df-hom 15898  df-prds 16040
This theorem is referenced by:  prdssca  16048  prdsbas  16049  prdsplusg  16050  prdsmulr  16051  prdsvsca  16052  prdsip  16053  prdsle  16054  prdsds  16056  prdstset  16058  prdshom  16059  prdsco  16060
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