MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isvclem Structured version   Visualization version   GIF version

Theorem isvclem 27278
Description: Lemma for isvcOLD 27280. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
isvclem.1 𝑋 = ran 𝐺
Assertion
Ref Expression
isvclem ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝑆,𝑦,𝑧   𝑥,𝑋,𝑧
Allowed substitution hint:   𝑋(𝑦)

Proof of Theorem isvclem
Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vc 27260 . . 3 CVecOLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))}
21eleq2i 2690 . 2 (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ ⟨𝐺, 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))})
3 eleq1 2686 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
4 rneq 5311 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
5 isvclem.1 . . . . . 6 𝑋 = ran 𝐺
64, 5syl6eqr 2673 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
7 xpeq2 5089 . . . . . . 7 (ran 𝑔 = 𝑋 → (ℂ × ran 𝑔) = (ℂ × 𝑋))
87feq2d 5988 . . . . . 6 (ran 𝑔 = 𝑋 → (𝑠:(ℂ × ran 𝑔)⟶ran 𝑔𝑠:(ℂ × 𝑋)⟶ran 𝑔))
9 feq3 5985 . . . . . 6 (ran 𝑔 = 𝑋 → (𝑠:(ℂ × 𝑋)⟶ran 𝑔𝑠:(ℂ × 𝑋)⟶𝑋))
108, 9bitrd 268 . . . . 5 (ran 𝑔 = 𝑋 → (𝑠:(ℂ × ran 𝑔)⟶ran 𝑔𝑠:(ℂ × 𝑋)⟶𝑋))
116, 10syl 17 . . . 4 (𝑔 = 𝐺 → (𝑠:(ℂ × ran 𝑔)⟶ran 𝑔𝑠:(ℂ × 𝑋)⟶𝑋))
12 oveq 6610 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑥𝑔𝑧) = (𝑥𝐺𝑧))
1312oveq2d 6620 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑠(𝑥𝑔𝑧)) = (𝑦𝑠(𝑥𝐺𝑧)))
14 oveq 6610 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)))
1513, 14eqeq12d 2636 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ↔ (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧))))
166, 15raleqbidv 3141 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ↔ ∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧))))
17 oveq 6610 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)))
1817eqeq2d 2631 . . . . . . . . . 10 (𝑔 = 𝐺 → (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ↔ ((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥))))
1918anbi1d 740 . . . . . . . . 9 (𝑔 = 𝐺 → ((((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))
2019ralbidv 2980 . . . . . . . 8 (𝑔 = 𝐺 → (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))
2116, 20anbi12d 746 . . . . . . 7 (𝑔 = 𝐺 → ((∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))) ↔ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))
2221ralbidv 2980 . . . . . 6 (𝑔 = 𝐺 → (∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))) ↔ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))
2322anbi2d 739 . . . . 5 (𝑔 = 𝐺 → (((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))) ↔ ((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))))
246, 23raleqbidv 3141 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))) ↔ ∀𝑥𝑋 ((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))))
253, 11, 243anbi123d 1396 . . 3 (𝑔 = 𝐺 → ((𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))) ↔ (𝐺 ∈ AbelOp ∧ 𝑠:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))))
26 feq1 5983 . . . 4 (𝑠 = 𝑆 → (𝑠:(ℂ × 𝑋)⟶𝑋𝑆:(ℂ × 𝑋)⟶𝑋))
27 oveq 6610 . . . . . . 7 (𝑠 = 𝑆 → (1𝑠𝑥) = (1𝑆𝑥))
2827eqeq1d 2623 . . . . . 6 (𝑠 = 𝑆 → ((1𝑠𝑥) = 𝑥 ↔ (1𝑆𝑥) = 𝑥))
29 oveq 6610 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑦𝑠(𝑥𝐺𝑧)) = (𝑦𝑆(𝑥𝐺𝑧)))
30 oveq 6610 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑦𝑠𝑥) = (𝑦𝑆𝑥))
31 oveq 6610 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑦𝑠𝑧) = (𝑦𝑆𝑧))
3230, 31oveq12d 6622 . . . . . . . . . 10 (𝑠 = 𝑆 → ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
3329, 32eqeq12d 2636 . . . . . . . . 9 (𝑠 = 𝑆 → ((𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ↔ (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧))))
3433ralbidv 2980 . . . . . . . 8 (𝑠 = 𝑆 → (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ↔ ∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧))))
35 oveq 6610 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑦 + 𝑧)𝑠𝑥) = ((𝑦 + 𝑧)𝑆𝑥))
36 oveq 6610 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑧𝑠𝑥) = (𝑧𝑆𝑥))
3730, 36oveq12d 6622 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
3835, 37eqeq12d 2636 . . . . . . . . . 10 (𝑠 = 𝑆 → (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ↔ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥))))
39 oveq 6610 . . . . . . . . . . 11 (𝑠 = 𝑆 → ((𝑦 · 𝑧)𝑠𝑥) = ((𝑦 · 𝑧)𝑆𝑥))
40 oveq 6610 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑦𝑠(𝑧𝑠𝑥)) = (𝑦𝑆(𝑧𝑠𝑥)))
4136oveq2d 6620 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑦𝑆(𝑧𝑠𝑥)) = (𝑦𝑆(𝑧𝑆𝑥)))
4240, 41eqtrd 2655 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑦𝑠(𝑧𝑠𝑥)) = (𝑦𝑆(𝑧𝑆𝑥)))
4339, 42eqeq12d 2636 . . . . . . . . . 10 (𝑠 = 𝑆 → (((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)) ↔ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
4438, 43anbi12d 746 . . . . . . . . 9 (𝑠 = 𝑆 → ((((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
4544ralbidv 2980 . . . . . . . 8 (𝑠 = 𝑆 → (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))) ↔ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
4634, 45anbi12d 746 . . . . . . 7 (𝑠 = 𝑆 → ((∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))) ↔ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))
4746ralbidv 2980 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))) ↔ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))
4828, 47anbi12d 746 . . . . 5 (𝑠 = 𝑆 → (((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))) ↔ ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
4948ralbidv 2980 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑋 ((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))) ↔ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
5026, 493anbi23d 1399 . . 3 (𝑠 = 𝑆 → ((𝐺 ∈ AbelOp ∧ 𝑠:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑠(𝑥𝐺𝑧)) = ((𝑦𝑠𝑥)𝐺(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝐺(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥)))))) ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))))
5125, 50opelopabg 4953 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (⟨𝐺, 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))} ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))))
522, 51syl5bb 272 1 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cop 4154  {copab 4672   × cxp 5072  ran crn 5075  wf 5843  (class class class)co 6604  cc 9878  1c1 9881   + caddc 9883   · cmul 9885  AbelOpcablo 27244  CVecOLDcvc 27259
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-vc 27260
This theorem is referenced by:  isvcOLD  27280
  Copyright terms: Public domain W3C validator