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Theorem ghomlinOLD 33658
Description: Obsolete version of ghmlin 17646 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 2620 . . . . 5 ran 𝐻 = ran 𝐻
31, 2elghomOLD 33657 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
43biimp3a 1430 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋⟶ran 𝐻 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
54simprd 479 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
6 fveq2 6178 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
76oveq1d 6650 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥)𝐻(𝐹𝑦)) = ((𝐹𝐴)𝐻(𝐹𝑦)))
8 oveq1 6642 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
98fveq2d 6182 . . . 4 (𝑥 = 𝐴 → (𝐹‘(𝑥𝐺𝑦)) = (𝐹‘(𝐴𝐺𝑦)))
107, 9eqeq12d 2635 . . 3 (𝑥 = 𝐴 → (((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)) ↔ ((𝐹𝐴)𝐻(𝐹𝑦)) = (𝐹‘(𝐴𝐺𝑦))))
11 fveq2 6178 . . . . 5 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1211oveq2d 6651 . . . 4 (𝑦 = 𝐵 → ((𝐹𝐴)𝐻(𝐹𝑦)) = ((𝐹𝐴)𝐻(𝐹𝐵)))
13 oveq2 6643 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1413fveq2d 6182 . . . 4 (𝑦 = 𝐵 → (𝐹‘(𝐴𝐺𝑦)) = (𝐹‘(𝐴𝐺𝐵)))
1512, 14eqeq12d 2635 . . 3 (𝑦 = 𝐵 → (((𝐹𝐴)𝐻(𝐹𝑦)) = (𝐹‘(𝐴𝐺𝑦)) ↔ ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
1610, 15rspc2v 3317 . 2 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)) → ((𝐹𝐴)𝐻(𝐹𝐵)) = (𝐹‘(𝐴𝐺𝐵))))
175, 16mpan9 486 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 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  ran crn 5105  wf 5872  cfv 5876  (class class class)co 6635  GrpOpcgr 27313   GrpOpHom cghomOLD 33653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-ghomOLD 33654
This theorem is referenced by:  ghomidOLD  33659  ghomdiv  33662
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