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Theorem ghomlinOLD 35783
Description: Obsolete version of ghmlin 18627 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)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵)))

Proof of Theorem ghomlinOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomlinOLD.1 . . . . 5 𝑋 = ran 𝐺
2 eqid 2737 . . . . 5 ran 𝐻 = ran 𝐻
31, 2elghomOLD 35782 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
43biimp3a 1471 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
54simprd 499 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
6 fveq2 6717 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
76oveq1d 7228 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥)𝐻(𝐹𝑦)) = ((𝐹𝐴)𝐻(𝐹𝑦)))
8 oveq1 7220 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
98fveq2d 6721 . . . 4 (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
107, 9eqeq12d 2753 . . 3 (𝑥 = 𝐴 → (((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)) ↔ ((𝐹𝐴)𝐻(𝐹𝑦)) = (𝐹‘(𝐴𝐺𝑦))))
11 fveq2 6717 . . . . 5 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1211oveq2d 7229 . . . 4 (𝑦 = 𝐵 → ((𝐹𝐴)𝐻(𝐹𝑦)) = ((𝐹𝐴)𝐻(𝐹𝐵)))
13 oveq2 7221 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1413fveq2d 6721 . . . 4 (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
1512, 14eqeq12d 2753 . . 3 (𝑦 = 𝐵 → (((𝐹𝐴)𝐻(𝐹𝑦)) = (𝐹‘(𝐴𝐺𝑦)) ↔ ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
1610, 15rspc2v 3547 . 2 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)) → ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
175, 16mpan9 510 1 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  ran crn 5552  wf 6376  cfv 6380  (class class class)co 7213  GrpOpcgr 28570   GrpOpHom cghomOLD 35778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-ghomOLD 35779
This theorem is referenced by:  ghomidOLD  35784  ghomdiv  35787
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