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Theorem ghomidOLD 37419
Description: Obsolete version of ghmid 19180 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 2725 . . . . . . 7 ran 𝐺 = ran 𝐺
2 ghomidOLD.1 . . . . . . 7 π‘ˆ = (GIdβ€˜πΊ)
31, 2grpoidcl 30368 . . . . . 6 (𝐺 ∈ GrpOp β†’ π‘ˆ ∈ ran 𝐺)
433ad2ant1 1130 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ π‘ˆ ∈ ran 𝐺)
54, 4jca 510 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (π‘ˆ ∈ ran 𝐺 ∧ π‘ˆ ∈ ran 𝐺))
61ghomlinOLD 37418 . . . 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 30370 . . . . . 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 6896 . . . 4 (𝐺 ∈ GrpOp β†’ (πΉβ€˜(π‘ˆπΊπ‘ˆ)) = (πΉβ€˜π‘ˆ))
11103ad2ant1 1130 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ (πΉβ€˜(π‘ˆπΊπ‘ˆ)) = (πΉβ€˜π‘ˆ))
127, 11eqtrd 2765 . 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 2725 . . . . . . 7 ran 𝐻 = ran 𝐻
141, 13elghomOLD 37417 . . . . . 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 1465 . . . . 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 493 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ 𝐹:ran 𝐺⟢ran 𝐻)
1716, 4ffvelcdmd 7090 . . 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 30374 . . . . 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 411 . . . 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 1131 . . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  ran crn 5673  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416  GrpOpcgr 30343  GIdcgi 30344   GrpOpHom cghomOLD 37413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-grpo 30347  df-gid 30348  df-ghomOLD 37414
This theorem is referenced by:  grpokerinj  37423  rngohom0  37502
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