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Theorem ghomdiv 35235
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 1189 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → 𝐻 ∈ GrpOp)
2 ghomdiv.1 . . . . . . 7 𝑋 = ran 𝐺
3 eqid 2824 . . . . . . 7 ran 𝐻 = ran 𝐻
42, 3ghomf 35233 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶ran 𝐻)
54ffvelrnda 6834 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐻)
65adantrr 716 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹𝐴) ∈ ran 𝐻)
74ffvelrnda 6834 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝐵𝑋) → (𝐹𝐵) ∈ ran 𝐻)
87adantrl 715 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹𝐵) ∈ ran 𝐻)
9 ghomdiv.3 . . . . 5 𝐶 = ( /𝑔𝐻)
103, 9grponpcan 28317 . . . 4 ((𝐻 ∈ GrpOp ∧ (𝐹𝐴) ∈ ran 𝐻 ∧ (𝐹𝐵) ∈ ran 𝐻) → (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)) = (𝐹𝐴))
111, 6, 8, 10syl3anc 1368 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)) = (𝐹𝐴))
12 ghomdiv.2 . . . . . . 7 𝐷 = ( /𝑔𝐺)
132, 12grponpcan 28317 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
14133expb 1117 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
15143ad2antl1 1182 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
1615fveq2d 6657 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = (𝐹𝐴))
172, 12grpodivcl 28313 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
18173expb 1117 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
19 simprr 772 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
2018, 19jca 515 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋))
21203ad2antl1 1182 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋))
222ghomlinOLD 35231 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)) = (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)))
2322eqcomd 2830 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ ((𝐴𝐷𝐵) ∈ 𝑋𝐵𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)))
2421, 23syldan 594 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘((𝐴𝐷𝐵)𝐺𝐵)) = ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)))
2511, 16, 243eqtr2rd 2866 . 2 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐹‘(𝐴𝐷𝐵))𝐻(𝐹𝐵)) = (((𝐹𝐴)𝐶(𝐹𝐵))𝐻(𝐹𝐵)))
26183ad2antl1 1182 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ∈ 𝑋)
274ffvelrnda 6834 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝐷𝐵) ∈ 𝑋) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
2826, 27syldan 594 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐷𝐵)) ∈ ran 𝐻)
293, 9grpodivcl 28313 . . . 4 ((𝐻 ∈ GrpOp ∧ (𝐹𝐴) ∈ ran 𝐻 ∧ (𝐹𝐵) ∈ ran 𝐻) → ((𝐹𝐴)𝐶(𝐹𝐵)) ∈ ran 𝐻)
301, 6, 8, 29syl3anc 1368 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐹𝐴)𝐶(𝐹𝐵)) ∈ ran 𝐻)
313grporcan 28292 . . 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 1369 . 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 235 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  ran crn 5539  cfv 6338  (class class class)co 7140  GrpOpcgr 28263   /𝑔 cgs 28266   GrpOpHom cghomOLD 35226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7674  df-2nd 7675  df-grpo 28267  df-gid 28268  df-ginv 28269  df-gdiv 28270  df-ghomOLD 35227
This theorem is referenced by:  grpokerinj  35236  rngohomsub  35316
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