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

Theorem isvclem 30334
Description: Lemma for isvcOLD 30336. (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 30316 . . 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 2819 . 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 2815 . . . 4 (𝑔 = 𝐺 β†’ (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
4 rneq 5928 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
5 isvclem.1 . . . . . 6 𝑋 = ran 𝐺
64, 5eqtr4di 2784 . . . . 5 (𝑔 = 𝐺 β†’ ran 𝑔 = 𝑋)
7 xpeq2 5690 . . . . . . 7 (ran 𝑔 = 𝑋 β†’ (β„‚ Γ— ran 𝑔) = (β„‚ Γ— 𝑋))
87feq2d 6696 . . . . . 6 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— ran 𝑔)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)⟢ran 𝑔))
9 feq3 6693 . . . . . 6 (ran 𝑔 = 𝑋 β†’ (𝑠:(β„‚ Γ— 𝑋)⟢ran 𝑔 ↔ 𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
108, 9bitrd 279 . . . . 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 7410 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (π‘₯𝑔𝑧) = (π‘₯𝐺𝑧))
1312oveq2d 7420 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (𝑦𝑠(π‘₯𝑔𝑧)) = (𝑦𝑠(π‘₯𝐺𝑧)))
14 oveq 7410 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)))
1513, 14eqeq12d 2742 . . . . . . . . 9 (𝑔 = 𝐺 β†’ ((𝑦𝑠(π‘₯𝑔𝑧)) = ((𝑦𝑠π‘₯)𝑔(𝑦𝑠𝑧)) ↔ (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧))))
166, 15raleqbidv 3336 . . . . . . . 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 7410 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ ((𝑦𝑠π‘₯)𝑔(𝑧𝑠π‘₯)) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)))
1817eqeq2d 2737 . . . . . . . . . 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 3171 . . . . . . . 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 3171 . . . . . 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 3336 . . . 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 6691 . . . 4 (𝑠 = 𝑆 β†’ (𝑠:(β„‚ Γ— 𝑋)βŸΆπ‘‹ ↔ 𝑆:(β„‚ Γ— 𝑋)βŸΆπ‘‹))
27 oveq 7410 . . . . . . 7 (𝑠 = 𝑆 β†’ (1𝑠π‘₯) = (1𝑆π‘₯))
2827eqeq1d 2728 . . . . . 6 (𝑠 = 𝑆 β†’ ((1𝑠π‘₯) = π‘₯ ↔ (1𝑆π‘₯) = π‘₯))
29 oveq 7410 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ (𝑦𝑠(π‘₯𝐺𝑧)) = (𝑦𝑆(π‘₯𝐺𝑧)))
30 oveq 7410 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (𝑦𝑠π‘₯) = (𝑦𝑆π‘₯))
31 oveq 7410 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (𝑦𝑠𝑧) = (𝑦𝑆𝑧))
3230, 31oveq12d 7422 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧)))
3329, 32eqeq12d 2742 . . . . . . . . 9 (𝑠 = 𝑆 β†’ ((𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ↔ (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧))))
3433ralbidv 3171 . . . . . . . 8 (𝑠 = 𝑆 β†’ (βˆ€π‘§ ∈ 𝑋 (𝑦𝑠(π‘₯𝐺𝑧)) = ((𝑦𝑠π‘₯)𝐺(𝑦𝑠𝑧)) ↔ βˆ€π‘§ ∈ 𝑋 (𝑦𝑆(π‘₯𝐺𝑧)) = ((𝑦𝑆π‘₯)𝐺(𝑦𝑆𝑧))))
35 oveq 7410 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ ((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦 + 𝑧)𝑆π‘₯))
36 oveq 7410 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑧𝑠π‘₯) = (𝑧𝑆π‘₯))
3730, 36oveq12d 7422 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯)))
3835, 37eqeq12d 2742 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ (((𝑦 + 𝑧)𝑠π‘₯) = ((𝑦𝑠π‘₯)𝐺(𝑧𝑠π‘₯)) ↔ ((𝑦 + 𝑧)𝑆π‘₯) = ((𝑦𝑆π‘₯)𝐺(𝑧𝑆π‘₯))))
39 oveq 7410 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ ((𝑦 Β· 𝑧)𝑠π‘₯) = ((𝑦 Β· 𝑧)𝑆π‘₯))
40 oveq 7410 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑦𝑠(𝑧𝑠π‘₯)) = (𝑦𝑆(𝑧𝑠π‘₯)))
4136oveq2d 7420 . . . . . . . . . . . 12 (𝑠 = 𝑆 β†’ (𝑦𝑆(𝑧𝑠π‘₯)) = (𝑦𝑆(𝑧𝑆π‘₯)))
4240, 41eqtrd 2766 . . . . . . . . . . 11 (𝑠 = 𝑆 β†’ (𝑦𝑠(𝑧𝑠π‘₯)) = (𝑦𝑆(𝑧𝑆π‘₯)))
4339, 42eqeq12d 2742 . . . . . . . . . 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 3171 . . . . . . . 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 3171 . . . . . 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 3171 . . . 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 5531 . 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 283 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468  βŸ¨cop 4629  {copab 5203   Γ— cxp 5667  ran crn 5670  βŸΆwf 6532  (class class class)co 7404  β„‚cc 11107  1c1 11110   + caddc 11112   Β· cmul 11114  AbelOpcablo 30301  CVecOLDcvc 30315
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  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-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-vc 30316
This theorem is referenced by:  isvcOLD  30336
  Copyright terms: Public domain W3C validator