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Theorem ghomlinOLD 36397
Description: Obsolete version of ghmlin 19021 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 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΉβ€˜π΄)𝐻(πΉβ€˜π΅)) = (πΉβ€˜(𝐴𝐺𝐡)))

Proof of Theorem ghomlinOLD
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomlinOLD.1 . . . . 5 𝑋 = ran 𝐺
2 eqid 2733 . . . . 5 ran 𝐻 = ran 𝐻
31, 2elghomOLD 36396 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:π‘‹βŸΆran 𝐻 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
43biimp3a 1470 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (𝐹:π‘‹βŸΆran 𝐻 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))))
54simprd 497 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))
6 fveq2 6846 . . . . 5 (π‘₯ = 𝐴 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π΄))
76oveq1d 7376 . . . 4 (π‘₯ = 𝐴 β†’ ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = ((πΉβ€˜π΄)𝐻(πΉβ€˜π‘¦)))
8 oveq1 7368 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯𝐺𝑦) = (𝐴𝐺𝑦))
98fveq2d 6850 . . . 4 (π‘₯ = 𝐴 β†’ (πΉβ€˜(π‘₯𝐺𝑦)) = (πΉβ€˜(𝐴𝐺𝑦)))
107, 9eqeq12d 2749 . . 3 (π‘₯ = 𝐴 β†’ (((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)) ↔ ((πΉβ€˜π΄)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(𝐴𝐺𝑦))))
11 fveq2 6846 . . . . 5 (𝑦 = 𝐡 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π΅))
1211oveq2d 7377 . . . 4 (𝑦 = 𝐡 β†’ ((πΉβ€˜π΄)𝐻(πΉβ€˜π‘¦)) = ((πΉβ€˜π΄)𝐻(πΉβ€˜π΅)))
13 oveq2 7369 . . . . 5 (𝑦 = 𝐡 β†’ (𝐴𝐺𝑦) = (𝐴𝐺𝐡))
1413fveq2d 6850 . . . 4 (𝑦 = 𝐡 β†’ (πΉβ€˜(𝐴𝐺𝑦)) = (πΉβ€˜(𝐴𝐺𝐡)))
1512, 14eqeq12d 2749 . . 3 (𝑦 = 𝐡 β†’ (((πΉβ€˜π΄)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(𝐴𝐺𝑦)) ↔ ((πΉβ€˜π΄)𝐻(πΉβ€˜π΅)) = (πΉβ€˜(𝐴𝐺𝐡))))
1610, 15rspc2v 3592 . 2 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)) β†’ ((πΉβ€˜π΄)𝐻(πΉβ€˜π΅)) = (πΉβ€˜(𝐴𝐺𝐡))))
175, 16mpan9 508 1 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΉβ€˜π΄)𝐻(πΉβ€˜π΅)) = (πΉβ€˜(𝐴𝐺𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  ran crn 5638  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  GrpOpcgr 29480   GrpOpHom cghomOLD 36392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-ghomOLD 36393
This theorem is referenced by:  ghomidOLD  36398  ghomdiv  36401
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