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Theorem ghomco 36759
Description: The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
ghomco (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) β†’ (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))
Proof of Theorem ghomco
Dummy variables 𝑒 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fco 6742 . . . . . . 7 ((𝑇:ran 𝐻⟢ran 𝐾 ∧ 𝑆:ran 𝐺⟢ran 𝐻) β†’ (𝑇 ∘ 𝑆):ran 𝐺⟢ran 𝐾)
21ancoms 460 . . . . . 6 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) β†’ (𝑇 ∘ 𝑆):ran 𝐺⟢ran 𝐾)
32ad2ant2r 746 . . . . 5 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) β†’ (𝑇 ∘ 𝑆):ran 𝐺⟢ran 𝐾)
43a1i 11 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ (((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) β†’ (𝑇 ∘ 𝑆):ran 𝐺⟢ran 𝐾))
5 ffvelcdm 7084 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ π‘₯ ∈ ran 𝐺) β†’ (π‘†β€˜π‘₯) ∈ ran 𝐻)
6 ffvelcdm 7084 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) β†’ (π‘†β€˜π‘¦) ∈ ran 𝐻)
75, 6anim12dan 620 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((π‘†β€˜π‘₯) ∈ ran 𝐻 ∧ (π‘†β€˜π‘¦) ∈ ran 𝐻))
8 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 = (π‘†β€˜π‘₯) β†’ (π‘‡β€˜π‘’) = (π‘‡β€˜(π‘†β€˜π‘₯)))
98oveq1d 7424 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = (π‘†β€˜π‘₯) β†’ ((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜π‘£)))
10 fvoveq1 7432 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = (π‘†β€˜π‘₯) β†’ (π‘‡β€˜(𝑒𝐻𝑣)) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻𝑣)))
119, 10eqeq12d 2749 . . . . . . . . . . . . . . . . . . . . . 22 (𝑒 = (π‘†β€˜π‘₯) β†’ (((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)) ↔ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻𝑣))))
12 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (π‘†β€˜π‘¦) β†’ (π‘‡β€˜π‘£) = (π‘‡β€˜(π‘†β€˜π‘¦)))
1312oveq2d 7425 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (π‘†β€˜π‘¦) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜π‘£)) = ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))))
14 oveq2 7417 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = (π‘†β€˜π‘¦) β†’ ((π‘†β€˜π‘₯)𝐻𝑣) = ((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)))
1514fveq2d 6896 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = (π‘†β€˜π‘¦) β†’ (π‘‡β€˜((π‘†β€˜π‘₯)𝐻𝑣)) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦))))
1613, 15eqeq12d 2749 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 = (π‘†β€˜π‘¦) β†’ (((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻𝑣)) ↔ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)))))
1711, 16rspc2va 3624 . . . . . . . . . . . . . . . . . . . . 21 ((((π‘†β€˜π‘₯) ∈ ran 𝐻 ∧ (π‘†β€˜π‘¦) ∈ ran 𝐻) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦))))
187, 17sylan 581 . . . . . . . . . . . . . . . . . . . 20 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦))))
1918an32s 651 . . . . . . . . . . . . . . . . . . 19 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦))))
2019adantllr 718 . . . . . . . . . . . . . . . . . 18 ((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦))))
2120adantllr 718 . . . . . . . . . . . . . . . . 17 (((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦))))
22 fveq2 6892 . . . . . . . . . . . . . . . . 17 (((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)) β†’ (π‘‡β€˜((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦))) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
2321, 22sylan9eq 2793 . . . . . . . . . . . . . . . 16 ((((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) ∧ ((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
2423anasss 468 . . . . . . . . . . . . . . 15 (((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ ((π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)))) β†’ ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
25 fvco3 6991 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ π‘₯ ∈ ran 𝐺) β†’ ((𝑇 ∘ 𝑆)β€˜π‘₯) = (π‘‡β€˜(π‘†β€˜π‘₯)))
2625ad2ant2r 746 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((𝑇 ∘ 𝑆)β€˜π‘₯) = (π‘‡β€˜(π‘†β€˜π‘₯)))
27 fvco3 6991 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑦 ∈ ran 𝐺) β†’ ((𝑇 ∘ 𝑆)β€˜π‘¦) = (π‘‡β€˜(π‘†β€˜π‘¦)))
2827ad2ant2rl 748 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((𝑇 ∘ 𝑆)β€˜π‘¦) = (π‘‡β€˜(π‘†β€˜π‘¦)))
2926, 28oveq12d 7427 . . . . . . . . . . . . . . . . 17 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ (((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))))
3029adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ (((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))))
3130ad2ant2r 746 . . . . . . . . . . . . . . 15 (((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ ((π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)))) β†’ (((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((π‘‡β€˜(π‘†β€˜π‘₯))𝐾(π‘‡β€˜(π‘†β€˜π‘¦))))
32 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 ran 𝐺 = ran 𝐺
3332grpocl 29753 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ GrpOp ∧ π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) β†’ (π‘₯𝐺𝑦) ∈ ran 𝐺)
34333expb 1121 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ GrpOp ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ (π‘₯𝐺𝑦) ∈ ran 𝐺)
35 fvco3 6991 . . . . . . . . . . . . . . . . . . 19 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ (π‘₯𝐺𝑦) ∈ ran 𝐺) β†’ ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
3635adantlr 714 . . . . . . . . . . . . . . . . . 18 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ (π‘₯𝐺𝑦) ∈ ran 𝐺) β†’ ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
3734, 36sylan2 594 . . . . . . . . . . . . . . . . 17 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ (𝐺 ∈ GrpOp ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺))) β†’ ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
3837anassrs 469 . . . . . . . . . . . . . . . 16 ((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
3938ad2ant2r 746 . . . . . . . . . . . . . . 15 (((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ ((π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)))) β†’ ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)) = (π‘‡β€˜(π‘†β€˜(π‘₯𝐺𝑦))))
4024, 31, 393eqtr4d 2783 . . . . . . . . . . . . . 14 (((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ ((π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺) ∧ ((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)))) β†’ (((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)))
4140expr 458 . . . . . . . . . . . . 13 (((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ (π‘₯ ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐺)) β†’ (((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)) β†’ (((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦))))
4241ralimdvva 3205 . . . . . . . . . . . 12 ((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ 𝐺 ∈ GrpOp) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)) β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦))))
4342an32s 651 . . . . . . . . . . 11 ((((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) ∧ 𝐺 ∈ GrpOp) β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)) β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦))))
4443ex 414 . . . . . . . . . 10 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) β†’ (𝐺 ∈ GrpOp β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)) β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)))))
4544com23 86 . . . . . . . . 9 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ 𝑇:ran 𝐻⟢ran 𝐾) ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))) β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)) β†’ (𝐺 ∈ GrpOp β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)))))
4645anasss 468 . . . . . . . 8 ((𝑆:ran 𝐺⟢ran 𝐻 ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)) β†’ (𝐺 ∈ GrpOp β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)))))
4746imp 408 . . . . . . 7 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) β†’ (𝐺 ∈ GrpOp β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦))))
4847an32s 651 . . . . . 6 (((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) β†’ (𝐺 ∈ GrpOp β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦))))
4948com12 32 . . . . 5 (𝐺 ∈ GrpOp β†’ (((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦))))
50493ad2ant1 1134 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ (((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦))))
514, 50jcad 514 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ (((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))) β†’ ((𝑇 ∘ 𝑆):ran 𝐺⟢ran 𝐾 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)))))
52 eqid 2733 . . . . . 6 ran 𝐻 = ran 𝐻
5332, 52elghomOLD 36755 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)))))
54533adant3 1133 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦)))))
55 eqid 2733 . . . . . 6 ran 𝐾 = ran 𝐾
5652, 55elghomOLD 36755 . . . . 5 ((𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ (𝑇 ∈ (𝐻 GrpOpHom 𝐾) ↔ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))))
57563adant1 1131 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ (𝑇 ∈ (𝐻 GrpOpHom 𝐾) ↔ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣)))))
5854, 57anbi12d 632 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) ↔ ((𝑆:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘†β€˜π‘₯)𝐻(π‘†β€˜π‘¦)) = (π‘†β€˜(π‘₯𝐺𝑦))) ∧ (𝑇:ran 𝐻⟢ran 𝐾 ∧ βˆ€π‘’ ∈ ran π»βˆ€π‘£ ∈ ran 𝐻((π‘‡β€˜π‘’)𝐾(π‘‡β€˜π‘£)) = (π‘‡β€˜(𝑒𝐻𝑣))))))
5932, 55elghomOLD 36755 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ ((𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾) ↔ ((𝑇 ∘ 𝑆):ran 𝐺⟢ran 𝐾 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)))))
60593adant2 1132 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ ((𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾) ↔ ((𝑇 ∘ 𝑆):ran 𝐺⟢ran 𝐾 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺(((𝑇 ∘ 𝑆)β€˜π‘₯)𝐾((𝑇 ∘ 𝑆)β€˜π‘¦)) = ((𝑇 ∘ 𝑆)β€˜(π‘₯𝐺𝑦)))))
6151, 58, 603imtr4d 294 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) β†’ ((𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾)) β†’ (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾)))
6261imp 408 1 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) β†’ (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  ran crn 5678   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  GrpOpcgr 29742   GrpOpHom cghomOLD 36751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-grpo 29746  df-ghomOLD 36752
This theorem is referenced by: (None)
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