Theorem ghomlinOLD 38209

Description: Obsolete version of ghmlin 19196 as of 15-Mar-2020. Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ghomlinOLD.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
ghomlinOLD	⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵)))

Proof of Theorem ghomlinOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghomlinOLD.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2		eqid 2736	. . . . 5 ⊢ ran 𝐻 = ran 𝐻
3	1, 2	elghomOLD 38208	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
4	3	biimp3a 1472	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
5	4	simprd 495	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
6		fveq2 6840	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
7	6	oveq1d 7382	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐻(𝐹‘𝑦)))
8		oveq1 7374	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
9	8	fveq2d 6844	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
10	7, 9	eqeq12d 2752	. . 3 ⊢ (𝑥 = 𝐴 → (((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)) ↔ ((𝐹‘𝐴)𝐻(𝐹‘𝑦)) = (𝐹‘(𝐴𝐺𝑦))))
11		fveq2 6840	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
12	11	oveq2d 7383	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐻(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐻(𝐹‘𝐵)))
13		oveq2 7375	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
14	13	fveq2d 6844	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
15	12, 14	eqeq12d 2752	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐹‘𝐴)𝐻(𝐹‘𝑦)) = (𝐹‘(𝐴𝐺𝑦)) ↔ ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
16	10, 15	rspc2v 3575	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)) → ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
17	5, 16	mpan9 506	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ran crn 5632  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  GrpOpcgr 30560   GrpOpHom cghomOLD 38204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-ghomOLD 38205

This theorem is referenced by:  ghomidOLD  38210  ghomdiv  38213