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Theorem prdsval 16313
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 16306 . . . 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 3339 . . . . . . . . . . . 12 𝑟 ∈ V
54rnex 7261 . . . . . . . . . . 11 ran 𝑟 ∈ V
65uniex 7114 . . . . . . . . . 10 ran 𝑟 ∈ V
76rnex 7261 . . . . . . . . 9 ran ran 𝑟 ∈ V
87uniex 7114 . . . . . . . 8 ran ran 𝑟 ∈ V
9 baseid 16117 . . . . . . . . . . . 12 Base = Slot (Base‘ndx)
109strfvss 16078 . . . . . . . . . . 11 (Base‘(𝑟𝑥)) ⊆ ran (𝑟𝑥)
11 fvssunirn 6374 . . . . . . . . . . . 12 (𝑟𝑥) ⊆ ran 𝑟
12 rnss 5505 . . . . . . . . . . . 12 ((𝑟𝑥) ⊆ ran 𝑟 → ran (𝑟𝑥) ⊆ ran ran 𝑟)
13 uniss 4606 . . . . . . . . . . . 12 (ran (𝑟𝑥) ⊆ ran ran 𝑟 ran (𝑟𝑥) ⊆ ran ran 𝑟)
1411, 12, 13mp2b 10 . . . . . . . . . . 11 ran (𝑟𝑥) ⊆ ran ran 𝑟
1510, 14sstri 3749 . . . . . . . . . 10 (Base‘(𝑟𝑥)) ⊆ ran ran 𝑟
1615rgenw 3058 . . . . . . . . 9 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ⊆ ran ran 𝑟
17 iunss 4709 . . . . . . . . 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 4951 . . . . . . 7 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∈ V
20 ixpssmap2g 8099 . . . . . . 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 6837 . . . . . . 7 ( 𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ↑𝑚 dom 𝑟) ∈ V
2322ssex 4950 . . . . . 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 479 . . . . . . . . 9 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
2625fveq1d 6350 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑟𝑥) = (𝑅𝑥))
2726fveq2d 6352 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Base‘(𝑟𝑥)) = (Base‘(𝑅𝑥)))
2827ixpeq2dv 8086 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥𝐼 (Base‘(𝑟𝑥)) = X𝑥𝐼 (Base‘(𝑅𝑥)))
2925dmeqd 5477 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑅)
30 prdsval.i . . . . . . . . 9 (𝜑 → dom 𝑅 = 𝐼)
3130ad2antrr 764 . . . . . . . 8 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑅 = 𝐼)
3229, 31eqtrd 2790 . . . . . . 7 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = 𝐼)
3332ixpeq1d 8082 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) = X𝑥𝐼 (Base‘(𝑟𝑥)))
34 prdsval.b . . . . . . 7 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
3534ad2antrr 764 . . . . . 6 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
3628, 33, 353eqtr4d 2800 . . . . 5 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) = 𝐵)
37 ovssunirn 6840 . . . . . . . . . . . . . . 15 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ran (Hom ‘(𝑟𝑥))
38 df-hom 16164 . . . . . . . . . . . . . . . . . 18 Hom = Slot 14
3938strfvss 16078 . . . . . . . . . . . . . . . . 17 (Hom ‘(𝑟𝑥)) ⊆ ran (𝑟𝑥)
4039, 14sstri 3749 . . . . . . . . . . . . . . . 16 (Hom ‘(𝑟𝑥)) ⊆ ran ran 𝑟
41 rnss 5505 . . . . . . . . . . . . . . . 16 ((Hom ‘(𝑟𝑥)) ⊆ ran ran 𝑟 → ran (Hom ‘(𝑟𝑥)) ⊆ ran ran ran 𝑟)
42 uniss 4606 . . . . . . . . . . . . . . . 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 3749 . . . . . . . . . . . . . 14 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑟
4544rgenw 3058 . . . . . . . . . . . . 13 𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ran ran ran 𝑟
46 ss2ixp 8083 . . . . . . . . . . . . 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 7260 . . . . . . . . . . . . 13 dom 𝑟 ∈ V
498rnex 7261 . . . . . . . . . . . . . 14 ran ran ran 𝑟 ∈ V
5049uniex 7114 . . . . . . . . . . . . 13 ran ran ran 𝑟 ∈ V
5148, 50ixpconst 8080 . . . . . . . . . . . 12 X𝑥 ∈ dom 𝑟 ran ran ran 𝑟 = ( ran ran ran 𝑟𝑚 dom 𝑟)
5247, 51sseqtri 3774 . . . . . . . . . . 11 X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ⊆ ( ran ran ran 𝑟𝑚 dom 𝑟)
53 ovex 6837 . . . . . . . . . . . 12 ( ran ran ran 𝑟𝑚 dom 𝑟) ∈ V
5453elpw2 4973 . . . . . . . . . . 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 3059 . . . . . . . . 9 𝑓𝑣𝑔𝑣 X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) ∈ 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟)
57 eqid 2756 . . . . . . . . . 10 (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) = (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)))
5857fmpt2 7401 . . . . . . . . 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 3339 . . . . . . . . 9 𝑣 ∈ V
6160, 60xpex 7123 . . . . . . . 8 (𝑣 × 𝑣) ∈ V
6253pwex 4993 . . . . . . . 8 𝒫 ( ran ran ran 𝑟𝑚 dom 𝑟) ∈ V
63 fex2 7282 . . . . . . . 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 1569 . . . . . . 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 479 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
6732adantr 472 . . . . . . . . . 10 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → dom 𝑟 = 𝐼)
6867ixpeq1d 8082 . . . . . . . . 9 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)))
6926fveq2d 6352 . . . . . . . . . . . 12 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Hom ‘(𝑟𝑥)) = (Hom ‘(𝑅𝑥)))
7069oveqd 6826 . . . . . . . . . . 11 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7170ixpeq2dv 8086 . . . . . . . . . 10 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7271adantr 472 . . . . . . . . 9 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7368, 72eqtrd 2790 . . . . . . . 8 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥)) = X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))
7466, 66, 73mpt2eq123dv 6878 . . . . . . 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 768 . . . . . . 7 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
7774, 76eqtr4d 2793 . . . . . 6 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) = 𝐻)
78 simplr 809 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑣 = 𝐵)
7978opeq2d 4556 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝐵⟩)
8026fveq2d 6352 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (+g‘(𝑟𝑥)) = (+g‘(𝑅𝑥)))
8180oveqd 6826 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))
8232, 81mpteq12dv 4881 . . . . . . . . . . . . . 14 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))
8382adantr 472 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))
8466, 66, 83mpt2eq123dv 6878 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
8584adantr 472 . . . . . . . . . . 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 771 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
8885, 87eqtr4d 2793 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥)))) = + )
8988opeq2d 4556 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩ = ⟨(+g‘ndx), + ⟩)
9026fveq2d 6352 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (.r‘(𝑟𝑥)) = (.r‘(𝑅𝑥)))
9190oveqd 6826 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))
9232, 91mpteq12dv 4881 . . . . . . . . . . . . . 14 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))
9392adantr 472 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))
9466, 66, 93mpt2eq123dv 6878 . . . . . . . . . . . 12 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥)))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
9594adantr 472 . . . . . . . . . . 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 771 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → × = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
9895, 97eqtr4d 2793 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥)))) = × )
9998opeq2d 4556 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩ = ⟨(.r‘ndx), × ⟩)
10079, 89, 99tpeq123d 4423 . . . . . . . 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 827 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑠 = 𝑆)
102101opeq2d 4556 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(Scalar‘ndx), 𝑠⟩ = ⟨(Scalar‘ndx), 𝑆⟩)
103 simpllr 817 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑠 = 𝑆)
104103fveq2d 6352 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = (Base‘𝑆))
105 prdsval.k . . . . . . . . . . . . . 14 𝐾 = (Base‘𝑆)
106104, 105syl6eqr 2808 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = 𝐾)
10726fveq2d 6352 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑟𝑥)) = ( ·𝑠 ‘(𝑅𝑥)))
108107oveqd 6826 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥)) = (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))
10932, 108mpteq12dv 4881 . . . . . . . . . . . . . 14 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))
110109adantr 472 . . . . . . . . . . . . 13 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))
111106, 66, 110mpt2eq123dv 6878 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 771 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → · = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
115112, 114eqtr4d 2793 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥)))) = · )
116115opeq2d 4556 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
11726fveq2d 6352 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (·𝑖‘(𝑟𝑥)) = (·𝑖‘(𝑅𝑥)))
118117oveqd 6826 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))
11932, 118mpteq12dv 4881 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))
120119adantr 472 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))
121103, 120oveq12d 6827 . . . . . . . . . . . . 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 6878 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 771 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → , = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
126123, 125eqtr4d 2793 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥))))) = , )
127126opeq2d 4556 . . . . . . . . 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 4423 . . . . . . . 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 3907 . . . . . . 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 817 . . . . . . . . . . . . 13 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑟 = 𝑅)
131130coeq2d 5436 . . . . . . . . . . . 12 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (TopOpen ∘ 𝑟) = (TopOpen ∘ 𝑅))
132131fveq2d 6352 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (∏t‘(TopOpen ∘ 𝑟)) = (∏t‘(TopOpen ∘ 𝑅)))
133 prdsval.o . . . . . . . . . . . 12 (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
134133ad4antr 771 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
135132, 134eqtr4d 2793 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (∏t‘(TopOpen ∘ 𝑟)) = 𝑂)
136135opeq2d 4556 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩ = ⟨(TopSet‘ndx), 𝑂⟩)
13766sseq2d 3770 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ({𝑓, 𝑔} ⊆ 𝑣 ↔ {𝑓, 𝑔} ⊆ 𝐵))
13826fveq2d 6352 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (le‘(𝑟𝑥)) = (le‘(𝑅𝑥)))
139138breqd 4811 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥) ↔ (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)))
14032, 139raleqbidv 3287 . . . . . . . . . . . . . . 15 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥) ↔ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)))
141140adantr 472 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥) ↔ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥)))
142137, 141anbi12d 749 . . . . . . . . . . . . 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 4864 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 771 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
147144, 146eqtr4d 2793 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))} = )
148147opeq2d 4556 . . . . . . . . 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 6352 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (dist‘(𝑟𝑥)) = (dist‘(𝑅𝑥)))
150149oveqd 6826 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥)) = ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥)))
15132, 150mpteq12dv 4881 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))))
152151adantr 472 . . . . . . . . . . . . . . . 16 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))))
153152rneqd 5504 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) = ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))))
154153uneq1d 3905 . . . . . . . . . . . . . 14 ((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}) = (ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}))
155154supeq1d 8513 . . . . . . . . . . . . 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 6878 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 771 . . . . . . . . . . 11 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → 𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
160157, 159eqtr4d 2793 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = 𝐷)
161160opeq2d 4556 . . . . . . . . 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 4423 . . . . . . . 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 479 . . . . . . . . . 10 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → = 𝐻)
164163opeq2d 4556 . . . . . . . . 9 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
16578sqxpeqd 5294 . . . . . . . . . . . 12 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑣 × 𝑣) = (𝐵 × 𝐵))
166163oveqd 6826 . . . . . . . . . . . . 13 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑐(2nd𝑎)) = (𝑐𝐻(2nd𝑎)))
167163fveq1d 6350 . . . . . . . . . . . . 13 (((((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ = 𝐻) → (𝑎) = (𝐻𝑎))
16826fveq2d 6352 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (comp‘(𝑟𝑥)) = (comp‘(𝑅𝑥)))
169168oveqd 6826 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥)) = (⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥)))
170169oveqd 6826 . . . . . . . . . . . . . . 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 4881 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . 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 6878 . . . . . . . . . . . 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 6878 . . . . . . . . . . 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 771 . . . . . . . . . . 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 2793 . . . . . . . . . 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 4556 . . . . . . . . 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 4416 . . . . . . . 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 3907 . . . . . . 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 3907 . . . . . 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 3698 . . . . 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 3698 . . . 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 682 . . 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 3348 . . . 4 (𝑆𝑊𝑆 ∈ V)
187185, 186syl 17 . . 3 (𝜑𝑆 ∈ V)
188 prdsval.r . . . 4 (𝜑𝑅𝑍)
189 elex 3348 . . . 4 (𝑅𝑍𝑅 ∈ V)
190188, 189syl 17 . . 3 (𝜑𝑅 ∈ V)
191 tpex 7118 . . . . . 6 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
192 tpex 7118 . . . . . 6 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
193191, 192unex 7117 . . . . 5 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V
194 tpex 7118 . . . . . 6 {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∈ V
195 prex 5054 . . . . . 6 {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩} ∈ V
196194, 195unex 7117 . . . . 5 ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ⟩}) ∈ V
197193, 196unex 7117 . . . 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 6949 . 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 2802 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 383   = wceq 1628  wcel 2135  wral 3046  Vcvv 3336  csb 3670  cun 3709  wss 3711  𝒫 cpw 4298  {csn 4317  {cpr 4319  {ctp 4321  cop 4323   cuni 4584   ciun 4668   class class class wbr 4800  {copab 4860  cmpt 4877   × cxp 5260  dom cdm 5262  ran crn 5263  ccom 5266  wf 6041  cfv 6045  (class class class)co 6809  cmpt2 6811  1st c1st 7327  2nd c2nd 7328  𝑚 cmap 8019  Xcixp 8070  supcsup 8507  0cc0 10124  1c1 10125  *cxr 10261   < clt 10262  4c4 11260  cdc 11681  ndxcnx 16052  Basecbs 16055  +gcplusg 16139  .rcmulr 16140  Scalarcsca 16142   ·𝑠 cvsca 16143  ·𝑖cip 16144  TopSetcts 16145  lecple 16146  distcds 16148  Hom chom 16150  compcco 16151  TopOpenctopn 16280  tcpt 16297   Σg cgsu 16299  Xscprds 16304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-i2m1 10192  ax-1ne0 10193  ax-rrecex 10196  ax-cnre 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-map 8021  df-ixp 8071  df-sup 8509  df-nn 11209  df-ndx 16058  df-slot 16059  df-base 16061  df-hom 16164  df-prds 16306
This theorem is referenced by:  prdssca  16314  prdsbas  16315  prdsplusg  16316  prdsmulr  16317  prdsvsca  16318  prdsip  16319  prdsle  16320  prdsds  16322  prdstset  16324  prdshom  16325  prdsco  16326
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