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Theorem ghomlinOLD 32857
Description: Obsolete version of ghmlin 17488 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 2610 . . . . 5 ran 𝐻 = ran 𝐻
31, 2elghomOLD 32856 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
43biimp3a 1424 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
54simprd 478 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
6 fveq2 6103 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
76oveq1d 6564 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥)𝐻(𝐹𝑦)) = ((𝐹𝐴)𝐻(𝐹𝑦)))
8 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
98fveq2d 6107 . . . 4 (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
107, 9eqeq12d 2625 . . 3 (𝑥 = 𝐴 → (((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)) ↔ ((𝐹𝐴)𝐻(𝐹𝑦)) = (𝐹‘(𝐴𝐺𝑦))))
11 fveq2 6103 . . . . 5 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1211oveq2d 6565 . . . 4 (𝑦 = 𝐵 → ((𝐹𝐴)𝐻(𝐹𝑦)) = ((𝐹𝐴)𝐻(𝐹𝐵)))
13 oveq2 6557 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1413fveq2d 6107 . . . 4 (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
1512, 14eqeq12d 2625 . . 3 (𝑦 = 𝐵 → (((𝐹𝐴)𝐻(𝐹𝑦)) = (𝐹‘(𝐴𝐺𝑦)) ↔ ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
1610, 15rspc2v 3293 . 2 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)) → ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
175, 16mpan9 485 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 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  ran crn 5039  wf 5800  cfv 5804  (class class class)co 6549  GrpOpcgr 26727   GrpOpHom cghomOLD 32852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ghomOLD 32853
This theorem is referenced by:  ghomidOLD  32858  ghomdiv  32861
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