Theorem ghomlinOLD 38394

Description: Obsolete version of ghmlin 19279 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 2765	. . . . 5 ⊢ ran 𝐻 = ran 𝐻
3	1, 2	elghomOLD 38393	. . . 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 1493	. . 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 500	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
6		fveq2 6871	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
7	6	oveq1d 7415	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐻(𝐹‘𝑦)))
8		oveq1 7407	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
9	8	fveq2d 6875	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
10	7, 9	eqeq12d 2781	. . 3 ⊢ (𝑥 = 𝐴 → (((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)) ↔ ((𝐹‘𝐴)𝐻(𝐹‘𝑦)) = (𝐹‘(𝐴𝐺𝑦))))
11		fveq2 6871	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
12	11	oveq2d 7416	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐻(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐻(𝐹‘𝐵)))
13		oveq2 7408	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
14	13	fveq2d 6875	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
15	12, 14	eqeq12d 2781	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐹‘𝐴)𝐻(𝐹‘𝑦)) = (𝐹‘(𝐴𝐺𝑦)) ↔ ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
16	10, 15	rspc2v 3595	. 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 515	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 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ran crn 5652  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  GrpOpcgr 30746   GrpOpHom cghomOLD 38389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-ghomOLD 38390

This theorem is referenced by:  ghomidOLD  38395  ghomdiv  38398