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Theorem crngohomfo 33437
 Description: The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
Hypotheses
Ref Expression
crnghomfo.1 𝐺 = (1st𝑅)
crnghomfo.2 𝑋 = ran 𝐺
crnghomfo.3 𝐽 = (1st𝑆)
crnghomfo.4 𝑌 = ran 𝐽
Assertion
Ref Expression
crngohomfo (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ CRingOps)

Proof of Theorem crngohomfo
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 791 . 2 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ RingOps)
2 foelrn 6334 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑦𝑌) → ∃𝑤𝑋 𝑦 = (𝐹𝑤))
32ex 450 . . . . . . 7 (𝐹:𝑋onto𝑌 → (𝑦𝑌 → ∃𝑤𝑋 𝑦 = (𝐹𝑤)))
4 foelrn 6334 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑧𝑌) → ∃𝑥𝑋 𝑧 = (𝐹𝑥))
54ex 450 . . . . . . 7 (𝐹:𝑋onto𝑌 → (𝑧𝑌 → ∃𝑥𝑋 𝑧 = (𝐹𝑥)))
63, 5anim12d 585 . . . . . 6 (𝐹:𝑋onto𝑌 → ((𝑦𝑌𝑧𝑌) → (∃𝑤𝑋 𝑦 = (𝐹𝑤) ∧ ∃𝑥𝑋 𝑧 = (𝐹𝑥))))
7 reeanv 3097 . . . . . 6 (∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) ↔ (∃𝑤𝑋 𝑦 = (𝐹𝑤) ∧ ∃𝑥𝑋 𝑧 = (𝐹𝑥)))
86, 7syl6ibr 242 . . . . 5 (𝐹:𝑋onto𝑌 → ((𝑦𝑌𝑧𝑌) → ∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥))))
98ad2antll 764 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑦𝑌𝑧𝑌) → ∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥))))
10 crnghomfo.1 . . . . . . . . . . . . . 14 𝐺 = (1st𝑅)
11 eqid 2621 . . . . . . . . . . . . . 14 (2nd𝑅) = (2nd𝑅)
12 crnghomfo.2 . . . . . . . . . . . . . 14 𝑋 = ran 𝐺
1310, 11, 12crngocom 33432 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ 𝑤𝑋𝑥𝑋) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
14133expb 1263 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑤𝑋𝑥𝑋)) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
15143ad2antl1 1221 . . . . . . . . . . 11 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
1615fveq2d 6152 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = (𝐹‘(𝑥(2nd𝑅)𝑤)))
17 crngorngo 33431 . . . . . . . . . . 11 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18 eqid 2621 . . . . . . . . . . . 12 (2nd𝑆) = (2nd𝑆)
1910, 12, 11, 18rngohommul 33401 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2017, 19syl3anl1 1371 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2110, 12, 11, 18rngohommul 33401 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥𝑋𝑤𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2221ancom2s 843 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2317, 22syl3anl1 1371 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2416, 20, 233eqtr3d 2663 . . . . . . . . 9 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
25 oveq12 6613 . . . . . . . . . 10 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
26 oveq12 6613 . . . . . . . . . . 11 ((𝑧 = (𝐹𝑥) ∧ 𝑦 = (𝐹𝑤)) → (𝑧(2nd𝑆)𝑦) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2726ancoms 469 . . . . . . . . . 10 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑧(2nd𝑆)𝑦) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2825, 27eqeq12d 2636 . . . . . . . . 9 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → ((𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦) ↔ ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤))))
2924, 28syl5ibrcom 237 . . . . . . . 8 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
3029ex 450 . . . . . . 7 ((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
31303expa 1262 . . . . . 6 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3231adantrr 752 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3332rexlimdvv 3030 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → (∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
349, 33syld 47 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑦𝑌𝑧𝑌) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
3534ralrimivv 2964 . 2 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ∀𝑦𝑌𝑧𝑌 (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))
36 crnghomfo.3 . . 3 𝐽 = (1st𝑆)
37 crnghomfo.4 . . 3 𝑌 = ran 𝐽
3836, 18, 37iscrngo2 33428 . 2 (𝑆 ∈ CRingOps ↔ (𝑆 ∈ RingOps ∧ ∀𝑦𝑌𝑧𝑌 (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
391, 35, 38sylanbrc 697 1 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ CRingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  ran crn 5075  –onto→wfo 5845  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  RingOpscrngo 33325   RngHom crnghom 33391  CRingOpsccring 33424 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-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-rngo 33326  df-rngohom 33394  df-com2 33421  df-crngo 33425 This theorem is referenced by: (None)
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