Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ghomdiv Structured version   Visualization version   GIF version
Theorem ghomdiv 37370
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 2727 . . . . . . 7 ran 𝐻 = ran 𝐻
42, 3ghomf 37368 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) β†’ 𝐹:π‘‹βŸΆran 𝐻)
54ffvelcdmda 7097 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴 ∈ 𝑋) β†’ (πΉβ€˜π΄) ∈ ran 𝐻)
65adantrr 715 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜π΄) ∈ ran 𝐻)
74ffvelcdmda 7097 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐡 ∈ 𝑋) β†’ (πΉβ€˜π΅) ∈ ran 𝐻)
87adantrl 714 . . . 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 30371 . . . 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 30371 . . . . . 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 6904 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜((𝐴𝐷𝐡)𝐺𝐡)) = (πΉβ€˜π΄))
172, 12grpodivcl 30367 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
18173expb 1117 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ∈ 𝑋)
19 simprr 771 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
2018, 19jca 510 . . . . 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 37366 . . . . 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 2733 . . . 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 589 . . 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 2774 . 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 7097 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐡) ∈ 𝑋) β†’ (πΉβ€˜(𝐴𝐷𝐡)) ∈ ran 𝐻)
2826, 27syldan 589 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (πΉβ€˜(𝐴𝐷𝐡)) ∈ ran 𝐻)
293, 9grpodivcl 30367 . . . 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 30346 . . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5681  β€˜cfv 6551  (class class class)co 7424  GrpOpcgr 30317   /𝑔 cgs 30320   GrpOpHom cghomOLD 37361
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-grpo 30321  df-gid 30322  df-ginv 30323  df-gdiv 30324  df-ghomOLD 37362
This theorem is referenced by:  grpokerinj  37371  rngohomsub  37451
  Copyright terms: Public domain W3C validator