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Theorem ghomidOLD 36752
Description: Obsolete version of ghmid 19097 as of 15-Mar-2020. A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ghomidOLD.1 π‘ˆ = (GIdβ€˜πΊ)
ghomidOLD.2 𝑇 = (GIdβ€˜π»)
Assertion
Ref Expression
ghomidOLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (πΉβ€˜π‘ˆ) = 𝑇)
Proof of Theorem ghomidOLD
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . . 7 ran 𝐺 = ran 𝐺
2 ghomidOLD.1 . . . . . . 7 π‘ˆ = (GIdβ€˜πΊ)
31, 2grpoidcl 29762 . . . . . 6 (𝐺 ∈ GrpOp β†’ π‘ˆ ∈ ran 𝐺)
433ad2ant1 1133 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ π‘ˆ ∈ ran 𝐺)
54, 4jca 512 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (π‘ˆ ∈ ran 𝐺 ∧ π‘ˆ ∈ ran 𝐺))
61ghomlinOLD 36751 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (π‘ˆ ∈ ran 𝐺 ∧ π‘ˆ ∈ ran 𝐺)) β†’ ((πΉβ€˜π‘ˆ)𝐻(πΉβ€˜π‘ˆ)) = (πΉβ€˜(π‘ˆπΊπ‘ˆ)))
75, 6mpdan 685 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ ((πΉβ€˜π‘ˆ)𝐻(πΉβ€˜π‘ˆ)) = (πΉβ€˜(π‘ˆπΊπ‘ˆ)))
81, 2grpolid 29764 . . . . . 6 ((𝐺 ∈ GrpOp ∧ π‘ˆ ∈ ran 𝐺) β†’ (π‘ˆπΊπ‘ˆ) = π‘ˆ)
93, 8mpdan 685 . . . . 5 (𝐺 ∈ GrpOp β†’ (π‘ˆπΊπ‘ˆ) = π‘ˆ)
109fveq2d 6895 . . . 4 (𝐺 ∈ GrpOp β†’ (πΉβ€˜(π‘ˆπΊπ‘ˆ)) = (πΉβ€˜π‘ˆ))
11103ad2ant1 1133 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (πΉβ€˜(π‘ˆπΊπ‘ˆ)) = (πΉβ€˜π‘ˆ))
127, 11eqtrd 2772 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ ((πΉβ€˜π‘ˆ)𝐻(πΉβ€˜π‘ˆ)) = (πΉβ€˜π‘ˆ))
13 eqid 2732 . . . . . . 7 ran 𝐻 = ran 𝐻
141, 13elghomOLD 36750 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
1514biimp3a 1469 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))))
1615simpld 495 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ 𝐹:ran 𝐺⟢ran 𝐻)
1716, 4ffvelcdmd 7087 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (πΉβ€˜π‘ˆ) ∈ ran 𝐻)
18 ghomidOLD.2 . . . . . 6 𝑇 = (GIdβ€˜π»)
1913, 18grpoid 29768 . . . . 5 ((𝐻 ∈ GrpOp ∧ (πΉβ€˜π‘ˆ) ∈ ran 𝐻) β†’ ((πΉβ€˜π‘ˆ) = 𝑇 ↔ ((πΉβ€˜π‘ˆ)𝐻(πΉβ€˜π‘ˆ)) = (πΉβ€˜π‘ˆ)))
2019ex 413 . . . 4 (𝐻 ∈ GrpOp β†’ ((πΉβ€˜π‘ˆ) ∈ ran 𝐻 β†’ ((πΉβ€˜π‘ˆ) = 𝑇 ↔ ((πΉβ€˜π‘ˆ)𝐻(πΉβ€˜π‘ˆ)) = (πΉβ€˜π‘ˆ))))
21203ad2ant2 1134 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ ((πΉβ€˜π‘ˆ) ∈ ran 𝐻 β†’ ((πΉβ€˜π‘ˆ) = 𝑇 ↔ ((πΉβ€˜π‘ˆ)𝐻(πΉβ€˜π‘ˆ)) = (πΉβ€˜π‘ˆ))))
2217, 21mpd 15 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ ((πΉβ€˜π‘ˆ) = 𝑇 ↔ ((πΉβ€˜π‘ˆ)𝐻(πΉβ€˜π‘ˆ)) = (πΉβ€˜π‘ˆ)))
2312, 22mpbird 256 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (πΉβ€˜π‘ˆ) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  GrpOpcgr 29737  GIdcgi 29738   GrpOpHom cghomOLD 36746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-grpo 29741  df-gid 29742  df-ghomOLD 36747
This theorem is referenced by:  grpokerinj  36756  rngohom0  36835
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