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 34045
Description: Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
ghomdiv.1 𝑋 = ran 𝐺
ghomdiv.2 𝐷 = ( /𝑔𝐺)
ghomdiv.3 𝐶 = ( /𝑔𝐻)
Assertion
Ref Expression
ghomdiv (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹𝐴)𝐶(𝐹𝐵)))

Proof of Theorem ghomdiv
StepHypRef Expression
1 simpl2 1244 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → 𝐻 ∈ GrpOp)
2 ghomdiv.1 . . . . . . 7 𝑋 = ran 𝐺
3 eqid 2765 . . . . . . 7 ran 𝐻 = ran 𝐻
42, 3ghomf 34043 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶ran 𝐻)
54ffvelrnda 6549 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐻)
65adantrr 708 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹𝐴) ∈ ran 𝐻)
74ffvelrnda 6549 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐵𝑋) → (𝐹𝐵) ∈ ran 𝐻)
87adantrl 707 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹𝐵) ∈ ran 𝐻)
9 ghomdiv.3 . . . . 5 𝐶 = ( /𝑔𝐻)
103, 9grponpcan 27789 . . . 4 ((𝐻 ∈ GrpOp ∧ (𝐹𝐴) ∈ ran 𝐻 ∧ (𝐹𝐵) ∈ ran 𝐻) → (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)) = (𝐹𝐴))
111, 6, 8, 10syl3anc 1490 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)) = (𝐹𝐴))
12 ghomdiv.2 . . . . . . 7 𝐷 = ( /𝑔𝐺)
132, 12grponpcan 27789 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
14133expb 1149 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
15143ad2antl1 1236 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
1615fveq2d 6379 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = (𝐹𝐴))
172, 12grpodivcl 27785 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
18173expb 1149 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
19 simprr 789 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
2018, 19jca 507 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋))
21203ad2antl1 1236 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋))
222ghomlinOLD 34041 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)) = (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)))
2322eqcomd 2771 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)))
2421, 23syldan 585 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)))
2511, 16, 243eqtr2rd 2806 . 2 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)) = (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)))
26183ad2antl1 1236 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
274ffvelrnda 6549 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐵) ∈ 𝑋) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
2826, 27syldan 585 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
293, 9grpodivcl 27785 . . . 4 ((𝐻 ∈ GrpOp ∧ (𝐹𝐴) ∈ ran 𝐻 ∧ (𝐹𝐵) ∈ ran 𝐻) → ((𝐹𝐴)𝐶(𝐹𝐵)) ∈ ran 𝐻)
301, 6, 8, 29syl3anc 1490 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐹𝐴)𝐶(𝐹𝐵)) ∈ ran 𝐻)
313grporcan 27764 . . 3 ((𝐻 ∈ GrpOp ∧ ((𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻 ∧ ((𝐹𝐴)𝐶(𝐹𝐵)) ∈ ran 𝐻 ∧ (𝐹𝐵) ∈ ran 𝐻)) → (((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)) = (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)) ↔ (𝐹‘(𝐴𝐷𝐵)) = ((𝐹𝐴)𝐶(𝐹𝐵))))
321, 28, 30, 8, 31syl13anc 1491 . 2 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)) = (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)) ↔ (𝐹‘(𝐴𝐷𝐵)) = ((𝐹𝐴)𝐶(𝐹𝐵))))
3325, 32mpbid 223 1 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹𝐴)𝐶(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  ran crn 5278  cfv 6068  (class class class)co 6842  GrpOpcgr 27735   /𝑔 cgs 27738   GrpOpHom cghomOLD 34036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-grpo 27739  df-gid 27740  df-ginv 27741  df-gdiv 27742  df-ghomOLD 34037
This theorem is referenced by:  grpokerinj  34046  rngohomsub  34126
  Copyright terms: Public domain W3C validator