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Theorem ghomdiv 37271
Description: Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
ghomdiv.1 𝑋 = ran 𝐺
ghomdiv.2 𝐷 = ( /𝑔 β€˜πΊ)
ghomdiv.3 𝐢 = ( /𝑔 β€˜π»)
Assertion
Ref Expression
ghomdiv (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐷𝐡)) = ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅)))
Proof of Theorem ghomdiv
StepHypRef Expression
1 simpl2 1189 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐻 ∈ GrpOp)
2 ghomdiv.1 . . . . . . 7 𝑋 = ran 𝐺
3 eqid 2726 . . . . . . 7 ran 𝐻 = ran 𝐻
42, 3ghomf 37269 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ 𝐹:π‘‹βŸΆran 𝐻)
54ffvelcdmda 7079 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐻)
65adantrr 714 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜π΄) ∈ ran 𝐻)
74ffvelcdmda 7079 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐡 ∈ 𝑋) β†’ (πΉβ€˜π΅) ∈ ran 𝐻)
87adantrl 713 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜π΅) ∈ ran 𝐻)
9 ghomdiv.3 . . . . 5 𝐢 = ( /𝑔 β€˜π»)
103, 9grponpcan 30301 . . . 4 ((𝐻 ∈ GrpOp ∧ (πΉβ€˜π΄) ∈ ran 𝐻 ∧ (πΉβ€˜π΅) ∈ ran 𝐻) β†’ (((πΉβ€˜π΄)𝐢(πΉβ€˜π΅))𝐻(πΉβ€˜π΅)) = (πΉβ€˜π΄))
111, 6, 8, 10syl3anc 1368 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (((πΉβ€˜π΄)𝐢(πΉβ€˜π΅))𝐻(πΉβ€˜π΅)) = (πΉβ€˜π΄))
12 ghomdiv.2 . . . . . . 7 𝐷 = ( /𝑔 β€˜πΊ)
132, 12grponpcan 30301 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
14133expb 1117 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
15143ad2antl1 1182 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
1615fveq2d 6888 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜((𝐴𝐷𝐡)𝐺𝐡)) = (πΉβ€˜π΄))
172, 12grpodivcl 30297 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
18173expb 1117 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
19 simprr 770 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
2018, 19jca 511 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
21203ad2antl1 1182 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
222ghomlinOLD 37267 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐡) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΉβ€˜(𝐴𝐷𝐡))𝐻(πΉβ€˜π΅)) = (πΉβ€˜((𝐴𝐷𝐡)𝐺𝐡)))
2322eqcomd 2732 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐡) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜((𝐴𝐷𝐡)𝐺𝐡)) = ((πΉβ€˜(𝐴𝐷𝐡))𝐻(πΉβ€˜π΅)))
2421, 23syldan 590 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜((𝐴𝐷𝐡)𝐺𝐡)) = ((πΉβ€˜(𝐴𝐷𝐡))𝐻(πΉβ€˜π΅)))
2511, 16, 243eqtr2rd 2773 . 2 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΉβ€˜(𝐴𝐷𝐡))𝐻(πΉβ€˜π΅)) = (((πΉβ€˜π΄)𝐢(πΉβ€˜π΅))𝐻(πΉβ€˜π΅)))
26183ad2antl1 1182 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
274ffvelcdmda 7079 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐡) ∈ 𝑋) β†’ (πΉβ€˜(𝐴𝐷𝐡)) ∈ ran 𝐻)
2826, 27syldan 590 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐷𝐡)) ∈ ran 𝐻)
293, 9grpodivcl 30297 . . . 4 ((𝐻 ∈ GrpOp ∧ (πΉβ€˜π΄) ∈ ran 𝐻 ∧ (πΉβ€˜π΅) ∈ ran 𝐻) β†’ ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅)) ∈ ran 𝐻)
301, 6, 8, 29syl3anc 1368 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅)) ∈ ran 𝐻)
313grporcan 30276 . . 3 ((𝐻 ∈ GrpOp ∧ ((πΉβ€˜(𝐴𝐷𝐡)) ∈ ran 𝐻 ∧ ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅)) ∈ ran 𝐻 ∧ (πΉβ€˜π΅) ∈ ran 𝐻)) β†’ (((πΉβ€˜(𝐴𝐷𝐡))𝐻(πΉβ€˜π΅)) = (((πΉβ€˜π΄)𝐢(πΉβ€˜π΅))𝐻(πΉβ€˜π΅)) ↔ (πΉβ€˜(𝐴𝐷𝐡)) = ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅))))
321, 28, 30, 8, 31syl13anc 1369 . 2 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (((πΉβ€˜(𝐴𝐷𝐡))𝐻(πΉβ€˜π΅)) = (((πΉβ€˜π΄)𝐢(πΉβ€˜π΅))𝐻(πΉβ€˜π΅)) ↔ (πΉβ€˜(𝐴𝐷𝐡)) = ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅))))
3325, 32mpbid 231 1 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐷𝐡)) = ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5670  β€˜cfv 6536  (class class class)co 7404  GrpOpcgr 30247   /𝑔 cgs 30250   GrpOpHom cghomOLD 37262
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-grpo 30251  df-gid 30252  df-ginv 30253  df-gdiv 30254  df-ghomOLD 37263
This theorem is referenced by:  grpokerinj  37272  rngohomsub  37352
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