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Theorem crngohomfo 38014
Description: The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
Hypotheses
Ref Expression
crngohomfo.1 𝐺 = (1st𝑅)
crngohomfo.2 𝑋 = ran 𝐺
crngohomfo.3 𝐽 = (1st𝑆)
crngohomfo.4 𝑌 = ran 𝐽
Assertion
Ref Expression
crngohomfo (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 768 . 2 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ RingOps)
2 foelrn 7126 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑦𝑌) → ∃𝑤𝑋 𝑦 = (𝐹𝑤))
32ex 412 . . . . . . 7 (𝐹:𝑋onto𝑌 → (𝑦𝑌 → ∃𝑤𝑋 𝑦 = (𝐹𝑤)))
4 foelrn 7126 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑧𝑌) → ∃𝑥𝑋 𝑧 = (𝐹𝑥))
54ex 412 . . . . . . 7 (𝐹:𝑋onto𝑌 → (𝑧𝑌 → ∃𝑥𝑋 𝑧 = (𝐹𝑥)))
63, 5anim12d 609 . . . . . 6 (𝐹:𝑋onto𝑌 → ((𝑦𝑌𝑧𝑌) → (∃𝑤𝑋 𝑦 = (𝐹𝑤) ∧ ∃𝑥𝑋 𝑧 = (𝐹𝑥))))
7 reeanv 3228 . . . . . 6 (∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) ↔ (∃𝑤𝑋 𝑦 = (𝐹𝑤) ∧ ∃𝑥𝑋 𝑧 = (𝐹𝑥)))
86, 7imbitrrdi 252 . . . . 5 (𝐹:𝑋onto𝑌 → ((𝑦𝑌𝑧𝑌) → ∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥))))
98ad2antll 729 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑦𝑌𝑧𝑌) → ∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥))))
10 crngohomfo.1 . . . . . . . . . . . . . 14 𝐺 = (1st𝑅)
11 eqid 2736 . . . . . . . . . . . . . 14 (2nd𝑅) = (2nd𝑅)
12 crngohomfo.2 . . . . . . . . . . . . . 14 𝑋 = ran 𝐺
1310, 11, 12crngocom 38009 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ 𝑤𝑋𝑥𝑋) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
14133expb 1120 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑤𝑋𝑥𝑋)) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
15143ad2antl1 1185 . . . . . . . . . . 11 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
1615fveq2d 6909 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = (𝐹‘(𝑥(2nd𝑅)𝑤)))
17 crngorngo 38008 . . . . . . . . . . 11 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18 eqid 2736 . . . . . . . . . . . 12 (2nd𝑆) = (2nd𝑆)
1910, 12, 11, 18rngohommul 37978 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2017, 19syl3anl1 1413 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2110, 12, 11, 18rngohommul 37978 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥𝑋𝑤𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2221ancom2s 650 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2317, 22syl3anl1 1413 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2416, 20, 233eqtr3d 2784 . . . . . . . . 9 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
25 oveq12 7441 . . . . . . . . . 10 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
26 oveq12 7441 . . . . . . . . . . 11 ((𝑧 = (𝐹𝑥) ∧ 𝑦 = (𝐹𝑤)) → (𝑧(2nd𝑆)𝑦) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2726ancoms 458 . . . . . . . . . 10 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑧(2nd𝑆)𝑦) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2825, 27eqeq12d 2752 . . . . . . . . 9 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → ((𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦) ↔ ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤))))
2924, 28syl5ibrcom 247 . . . . . . . 8 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
3029ex 412 . . . . . . 7 ((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
31303expa 1118 . . . . . 6 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3231adantrr 717 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3332rexlimdvv 3211 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → (∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
349, 33syld 47 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑦𝑌𝑧𝑌) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
3534ralrimivv 3199 . 2 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ∀𝑦𝑌𝑧𝑌 (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))
36 crngohomfo.3 . . 3 𝐽 = (1st𝑆)
37 crngohomfo.4 . . 3 𝑌 = ran 𝐽
3836, 18, 37iscrngo2 38005 . 2 (𝑆 ∈ CRingOps ↔ (𝑆 ∈ RingOps ∧ ∀𝑦𝑌𝑧𝑌 (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
391, 35, 38sylanbrc 583 1 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  ran crn 5685  ontowfo 6558  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  RingOpscrngo 37902   RingOpsHom crngohom 37968  CRingOpsccring 38001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-rngo 37903  df-rngohom 37971  df-com2 37998  df-crngo 38002
This theorem is referenced by: (None)
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