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Theorem rngohomsub 33443
Description: Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
Hypotheses
Ref Expression
rnghomsub.1 𝐺 = (1st𝑅)
rnghomsub.2 𝑋 = ran 𝐺
rnghomsub.3 𝐻 = ( /𝑔𝐺)
rnghomsub.4 𝐽 = (1st𝑆)
rnghomsub.5 𝐾 = ( /𝑔𝐽)
Assertion
Ref Expression
rngohomsub (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹𝐴)𝐾(𝐹𝐵)))

Proof of Theorem rngohomsub
StepHypRef Expression
1 rnghomsub.1 . . . . 5 𝐺 = (1st𝑅)
21rngogrpo 33380 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
323ad2ant1 1080 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp)
4 rnghomsub.4 . . . . 5 𝐽 = (1st𝑆)
54rngogrpo 33380 . . . 4 (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
653ad2ant2 1081 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp)
71, 4rngogrphom 33441 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
83, 6, 73jca 1240 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)))
9 rnghomsub.2 . . 3 𝑋 = ran 𝐺
10 rnghomsub.3 . . 3 𝐻 = ( /𝑔𝐺)
11 rnghomsub.5 . . 3 𝐾 = ( /𝑔𝐽)
129, 10, 11ghomdiv 33362 . 2 (((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹𝐴)𝐾(𝐹𝐵)))
138, 12sylan 488 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹𝐴)𝐾(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ran crn 5085  cfv 5857  (class class class)co 6615  1st c1st 7126  GrpOpcgr 27231   /𝑔 cgs 27234   GrpOpHom cghomOLD 33353  RingOpscrngo 33364   RngHom crnghom 33430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819  df-grpo 27235  df-gid 27236  df-ginv 27237  df-gdiv 27238  df-ablo 27287  df-ghomOLD 33354  df-rngo 33365  df-rngohom 33433
This theorem is referenced by: (None)
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