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Theorem vciOLD 30319
Description: Obsolete version of cvsi 25008. The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable π‘Š was chosen because V is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
vciOLD.1 𝐺 = (1st β€˜π‘Š)
vciOLD.2 𝑆 = (2nd β€˜π‘Š)
vciOLD.3 𝑋 = ran 𝐺
Assertion
Ref Expression
vciOLD (π‘Š ∈ CVecOLD 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‘₯)))))))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐺   π‘₯,𝑆,𝑦,𝑧   π‘₯,𝑋,𝑦,𝑧
Allowed substitution hints:   π‘Š(π‘₯,𝑦,𝑧)

Proof of Theorem vciOLD
Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vciOLD.1 . . . . 5 𝐺 = (1st β€˜π‘Š)
21eqeq2i 2739 . . . 4 (𝑔 = 𝐺 ↔ 𝑔 = (1st β€˜π‘Š))
3 eleq1 2815 . . . . 5 (𝑔 = 𝐺 β†’ (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
4 rneq 5928 . . . . . . 7 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
5 vciOLD.3 . . . . . . 7 𝑋 = ran 𝐺
64, 5eqtr4di 2784 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = 𝑋)
7 xpeq2 5690 . . . . . . . 8 (ran 𝑔 = 𝑋 β†’ (β„‚ Γ— ran 𝑔) = (β„‚ Γ— 𝑋))
87feq2d 6696 . . . . . . 7 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)⟢ran 𝑔))
9 feq3 6693 . . . . . . 7 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— 𝑋)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
108, 9bitrd 279 . . . . . 6 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
116, 10syl 17 . . . . 5 (𝑔 = 𝐺 β†’ (𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
12 oveq 7410 . . . . . . . . . . . 12 (𝑔 = 𝐺 β†’ (π‘₯𝑔𝑧) = (π‘₯𝐺𝑧))
1312oveq2d 7420 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (𝑦𝑠(π‘₯𝑔𝑧)) = (𝑦𝑠(π‘₯𝐺𝑧)))
14 oveq 7410 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)))
1513, 14eqeq12d 2742 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ ((𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ↔ (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧))))
166, 15raleqbidv 3336 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (βˆ€π‘§ ∈ ran 𝑔(𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ↔ βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧))))
17 oveq 7410 . . . . . . . . . . . 12 (𝑔 = 𝐺 β†’ ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)))
1817eqeq2d 2737 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) ↔ ((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯))))
1918anbi1d 629 . . . . . . . . . 10 (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 3171 . . . . . . . . 9 (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 . . . . . . . 8 (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 3171 . . . . . . 7 (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 . . . . . 6 (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 3336 . . . . 5 (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 . . . 4 (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‘₯))))))))
262, 25sylbir 234 . . 3 (𝑔 = (1st β€˜π‘Š) 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‘₯))))))))
27 vciOLD.2 . . . . 5 𝑆 = (2nd β€˜π‘Š)
2827eqeq2i 2739 . . . 4 (𝑠 = 𝑆 ↔ 𝑠 = (2nd β€˜π‘Š))
29 feq1 6691 . . . . 5 (𝑠 = 𝑆 β†’ (𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ↔ 𝑆:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
30 oveq 7410 . . . . . . . 8 (𝑠 = 𝑆 β†’ (1𝑠π‘₯) = (1𝑆π‘₯))
3130eqeq1d 2728 . . . . . . 7 (𝑠 = 𝑆 β†’ ((1𝑠π‘₯) = π‘₯ ↔ (1𝑆π‘₯) = π‘₯))
32 oveq 7410 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (𝑦𝑠(π‘₯𝐺𝑧)) = (𝑦𝑆(π‘₯𝐺𝑧)))
33 oveq 7410 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑦𝑠π‘₯) = (𝑦𝑆π‘₯))
34 oveq 7410 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑦𝑠𝑧) = (𝑦𝑆𝑧))
3533, 34oveq12d 7422 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)))
3632, 35eqeq12d 2742 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ ((𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ↔ (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧))))
3736ralbidv 3171 . . . . . . . . 9 (𝑠 = 𝑆 β†’ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ↔ βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧))))
38 oveq 7410 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ ((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦 + 𝑧)𝑆π‘₯))
39 oveq 7410 . . . . . . . . . . . . 13 (𝑠 = 𝑆 β†’ (𝑧𝑠π‘₯) = (𝑧𝑆π‘₯))
4033, 39oveq12d 7422 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)))
4138, 40eqeq12d 2742 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ↔ ((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯))))
42 oveq 7410 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ ((𝑦 Β· 𝑧)𝑠π‘₯) = ((𝑦 Β· 𝑧)𝑆π‘₯))
4339oveq2d 7420 . . . . . . . . . . . . 13 (𝑠 = 𝑆 β†’ (𝑦𝑠(𝑧𝑠π‘₯)) = (𝑦𝑠(𝑧𝑆π‘₯)))
44 oveq 7410 . . . . . . . . . . . . 13 (𝑠 = 𝑆 β†’ (𝑦𝑠(𝑧𝑆π‘₯)) = (𝑦𝑆(𝑧𝑆π‘₯)))
4543, 44eqtrd 2766 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑦𝑠(𝑧𝑠π‘₯)) = (𝑦𝑆(𝑧𝑆π‘₯)))
4642, 45eqeq12d 2742 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (((𝑦 Β· 𝑧)𝑠π‘₯) = (𝑦𝑠(𝑧𝑠π‘₯)) ↔ ((𝑦 Β· 𝑧)𝑆π‘₯) = (𝑦𝑆(𝑧𝑆π‘₯))))
4741, 46anbi12d 630 . . . . . . . . . 10 (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‘₯)))))
4847ralbidv 3171 . . . . . . . . 9 (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‘₯)))))
4937, 48anbi12d 630 . . . . . . . 8 (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‘₯))))))
5049ralbidv 3171 . . . . . . 7 (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‘₯))))))
5131, 50anbi12d 630 . . . . . 6 (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‘₯)))))))
5251ralbidv 3171 . . . . 5 (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‘₯)))))))
5329, 523anbi23d 1435 . . . 4 (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‘₯))))))))
5428, 53sylbir 234 . . 3 (𝑠 = (2nd β€˜π‘Š) 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‘₯))))))))
5526, 54elopabi 8044 . 2 (π‘Š ∈ {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‘₯))))))} 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‘₯)))))))
56 df-vc 30317 . 2 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‘₯))))))}
5755, 56eleq2s 2845 1 (π‘Š ∈ CVecOLD 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‘₯)))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {copab 5203   Γ— cxp 5667  ran crn 5670  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  β„‚cc 11107  1c1 11110   + caddc 11112   Β· cmul 11114  AbelOpcablo 30302  CVecOLDcvc 30316
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-1st 7971  df-2nd 7972  df-vc 30317
This theorem is referenced by:  vcsm  30320  vcidOLD  30322  vcdi  30323  vcdir  30324  vcass  30325  vcablo  30327
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