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Theorem isvclem 30407
Description: Lemma for isvcOLD 30409. (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) β†’ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 ((1𝑆π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))))))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐺   π‘₯,𝑆,𝑦,𝑧   π‘₯,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 30389 . . 3 CVecOLD = {βŸ¨π‘”, π‘ βŸ© ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ∧ βˆ€π‘₯ ∈ ran 𝑔((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))))}
21eleq2i 2821 . 2 (⟨𝐺, π‘†βŸ© ∈ CVecOLD ↔ ⟨𝐺, π‘†βŸ© ∈ {βŸ¨π‘”, π‘ βŸ© ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ∧ βˆ€π‘₯ ∈ ran 𝑔((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))))})
3 eleq1 2817 . . . 4 (𝑔 = 𝐺 β†’ (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
4 rneq 5942 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
5 isvclem.1 . . . . . 6 𝑋 = ran 𝐺
64, 5eqtr4di 2786 . . . . 5 (𝑔 = 𝐺 β†’ ran 𝑔 = 𝑋)
7 xpeq2 5703 . . . . . . 7 (ran 𝑔 = 𝑋 β†’ (β„‚ Γ— ran 𝑔) = (β„‚ Γ— 𝑋))
87feq2d 6713 . . . . . 6 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)⟢ran 𝑔))
9 feq3 6710 . . . . . 6 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— 𝑋)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
108, 9bitrd 278 . . . . 5 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
116, 10syl 17 . . . 4 (𝑔 = 𝐺 β†’ (𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
12 oveq 7432 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (π‘₯𝑔𝑧) = (π‘₯𝐺𝑧))
1312oveq2d 7442 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (𝑦𝑠(π‘₯𝑔𝑧)) = (𝑦𝑠(π‘₯𝐺𝑧)))
14 oveq 7432 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)))
1513, 14eqeq12d 2744 . . . . . . . . 9 (𝑔 = 𝐺 β†’ ((𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ↔ (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧))))
166, 15raleqbidv 3340 . . . . . . . 8 (𝑔 = 𝐺 β†’ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ↔ βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧))))
17 oveq 7432 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)))
1817eqeq2d 2739 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ↔ ((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯))))
1918anbi1d 629 . . . . . . . . 9 (𝑔 = 𝐺 β†’ ((((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))) ↔ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))))
2019ralbidv 3175 . . . . . . . 8 (𝑔 = 𝐺 β†’ (βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))) ↔ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))))
2116, 20anbi12d 630 . . . . . . 7 (𝑔 = 𝐺 β†’ ((βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))) ↔ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))))
2221ralbidv 3175 . . . . . 6 (𝑔 = 𝐺 β†’ (βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))) ↔ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))))
2322anbi2d 628 . . . . 5 (𝑔 = 𝐺 β†’ (((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))) ↔ ((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))))))
246, 23raleqbidv 3340 . . . 4 (𝑔 = 𝐺 β†’ (βˆ€π‘₯ ∈ ran 𝑔((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))) ↔ βˆ€π‘₯ ∈ 𝑋 ((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))))))
253, 11, 243anbi123d 1432 . . 3 (𝑔 = 𝐺 β†’ ((𝑔 ∈ AbelOp ∧ 𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ∧ βˆ€π‘₯ ∈ ran 𝑔((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))))) ↔ (𝐺 ∈ AbelOp ∧ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 ((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))))))
26 feq1 6708 . . . 4 (𝑠 = 𝑆 β†’ (𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ↔ 𝑆:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
27 oveq 7432 . . . . . . 7 (𝑠 = 𝑆 β†’ (1𝑠π‘₯) = (1𝑆π‘₯))
2827eqeq1d 2730 . . . . . 6 (𝑠 = 𝑆 β†’ ((1𝑠π‘₯) = π‘₯ ↔ (1𝑆π‘₯) = π‘₯))
29 oveq 7432 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ (𝑦𝑠(π‘₯𝐺𝑧)) = (𝑦𝑆(π‘₯𝐺𝑧)))
30 oveq 7432 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (𝑦𝑠π‘₯) = (𝑦𝑆π‘₯))
31 oveq 7432 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (𝑦𝑠𝑧) = (𝑦𝑆𝑧))
3230, 31oveq12d 7444 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)))
3329, 32eqeq12d 2744 . . . . . . . . 9 (𝑠 = 𝑆 β†’ ((𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ↔ (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧))))
3433ralbidv 3175 . . . . . . . 8 (𝑠 = 𝑆 β†’ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ↔ βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧))))
35 oveq 7432 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ ((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦 + 𝑧)𝑆π‘₯))
36 oveq 7432 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑧𝑠π‘₯) = (𝑧𝑆π‘₯))
3730, 36oveq12d 7444 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)))
3835, 37eqeq12d 2744 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ↔ ((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯))))
39 oveq 7432 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ ((𝑦 Β· 𝑧)𝑠π‘₯) = ((𝑦 Β· 𝑧)𝑆π‘₯))
40 oveq 7432 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑦𝑠(𝑧𝑠π‘₯)) = (𝑦𝑆(𝑧𝑠π‘₯)))
4136oveq2d 7442 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑦𝑆(𝑧𝑠π‘₯)) = (𝑦𝑆(𝑧𝑆π‘₯)))
4240, 41eqtrd 2768 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (𝑦𝑠(𝑧𝑠π‘₯)) = (𝑦𝑆(𝑧𝑆π‘₯)))
4339, 42eqeq12d 2744 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ (((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)) ↔ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))
4438, 43anbi12d 630 . . . . . . . . 9 (𝑠 = 𝑆 β†’ ((((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))) ↔ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯)))))
4544ralbidv 3175 . . . . . . . 8 (𝑠 = 𝑆 β†’ (βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))) ↔ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯)))))
4634, 45anbi12d 630 . . . . . . 7 (𝑠 = 𝑆 β†’ ((βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))) ↔ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))))
4746ralbidv 3175 . . . . . 6 (𝑠 = 𝑆 β†’ (βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))) ↔ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))))
4828, 47anbi12d 630 . . . . 5 (𝑠 = 𝑆 β†’ (((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))) ↔ ((1𝑆π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯)))))))
4948ralbidv 3175 . . . 4 (𝑠 = 𝑆 β†’ (βˆ€π‘₯ ∈ 𝑋 ((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))) ↔ βˆ€π‘₯ ∈ 𝑋 ((1𝑆π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯)))))))
5026, 493anbi23d 1435 . . 3 (𝑠 = 𝑆 β†’ ((𝐺 ∈ AbelOp ∧ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 ((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)))))) ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 ((1𝑆π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))))))
5125, 50opelopabg 5544 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V) β†’ (⟨𝐺, π‘†βŸ© ∈ {βŸ¨π‘”, π‘ βŸ© ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ∧ βˆ€π‘₯ ∈ ran 𝑔((1𝑠π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯))))))} ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 ((1𝑆π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))))))
522, 51bitrid 282 1 ((𝐺 ∈ V ∧ 𝑆 ∈ V) β†’ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 ((1𝑆π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)) ∧ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  Vcvv 3473  βŸ¨cop 4638  {copab 5214   Γ— cxp 5680  ran crn 5683  βŸΆwf 6549  (class class class)co 7426  β„‚cc 11144  1c1 11147   + caddc 11149   Β· cmul 11151  AbelOpcablo 30374  CVecOLDcvc 30388
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-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-vc 30389
This theorem is referenced by:  isvcOLD  30409
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