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Theorem rngohomsub 35404
 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 35341 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
323ad2ant1 1130 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐺 ∈ GrpOp)
4 rnghomsub.4 . . . . 5 𝐽 = (1st𝑆)
54rngogrpo 35341 . . . 4 (𝑆 ∈ RingOps → 𝐽 ∈ GrpOp)
653ad2ant2 1131 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐽 ∈ GrpOp)
71, 4rngogrphom 35402 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))
83, 6, 73jca 1125 . 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 35323 . 2 (((𝐺 ∈ GrpOp ∧ 𝐽 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐽)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹𝐴)𝐾(𝐹𝐵)))
138, 12sylan 583 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ran crn 5524  ‘cfv 6328  (class class class)co 7139  1st c1st 7673  GrpOpcgr 28275   /𝑔 cgs 28278   GrpOpHom cghomOLD 35314  RingOpscrngo 35325   RngHom crnghom 35391 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-grpo 28279  df-gid 28280  df-ginv 28281  df-gdiv 28282  df-ablo 28331  df-ghomOLD 35315  df-rngo 35326  df-rngohom 35394 This theorem is referenced by: (None)
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