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Theorem ghomdiv 36401
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 1193 . . . 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 2733 . . . . . . 7 ran 𝐻 = ran 𝐻
42, 3ghomf 36399 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ 𝐹:π‘‹βŸΆran 𝐻)
54ffvelcdmda 7039 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐻)
65adantrr 716 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜π΄) ∈ ran 𝐻)
74ffvelcdmda 7039 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐡 ∈ 𝑋) β†’ (πΉβ€˜π΅) ∈ ran 𝐻)
87adantrl 715 . . . 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 29534 . . . 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 1372 . . 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 29534 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
14133expb 1121 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
15143ad2antl1 1186 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐺𝐡) = 𝐴)
1615fveq2d 6850 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜((𝐴𝐷𝐡)𝐺𝐡)) = (πΉβ€˜π΄))
172, 12grpodivcl 29530 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
18173expb 1121 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
19 simprr 772 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
2018, 19jca 513 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
21203ad2antl1 1186 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡) ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
222ghomlinOLD 36397 . . . . 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 2739 . . . 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 592 . . 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 2780 . 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 1186 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
274ffvelcdmda 7039 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐡) ∈ 𝑋) β†’ (πΉβ€˜(𝐴𝐷𝐡)) ∈ ran 𝐻)
2826, 27syldan 592 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐷𝐡)) ∈ ran 𝐻)
293, 9grpodivcl 29530 . . . 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 1372 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΉβ€˜π΄)𝐢(πΉβ€˜π΅)) ∈ ran 𝐻)
313grporcan 29509 . . 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 1373 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ran crn 5638  β€˜cfv 6500  (class class class)co 7361  GrpOpcgr 29480   /𝑔 cgs 29483   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-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-grpo 29484  df-gid 29485  df-ginv 29486  df-gdiv 29487  df-ghomOLD 36393
This theorem is referenced by:  grpokerinj  36402  rngohomsub  36482
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