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Theorem crngohomfo 36224
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 766 . 2 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ RingOps)
2 foelrn 7019 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑦𝑌) → ∃𝑤𝑋 𝑦 = (𝐹𝑤))
32ex 413 . . . . . . 7 (𝐹:𝑋onto𝑌 → (𝑦𝑌 → ∃𝑤𝑋 𝑦 = (𝐹𝑤)))
4 foelrn 7019 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑧𝑌) → ∃𝑥𝑋 𝑧 = (𝐹𝑥))
54ex 413 . . . . . . 7 (𝐹:𝑋onto𝑌 → (𝑧𝑌 → ∃𝑥𝑋 𝑧 = (𝐹𝑥)))
63, 5anim12d 609 . . . . . 6 (𝐹:𝑋onto𝑌 → ((𝑦𝑌𝑧𝑌) → (∃𝑤𝑋 𝑦 = (𝐹𝑤) ∧ ∃𝑥𝑋 𝑧 = (𝐹𝑥))))
7 reeanv 3214 . . . . . 6 (∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) ↔ (∃𝑤𝑋 𝑦 = (𝐹𝑤) ∧ ∃𝑥𝑋 𝑧 = (𝐹𝑥)))
86, 7syl6ibr 251 . . . . 5 (𝐹:𝑋onto𝑌 → ((𝑦𝑌𝑧𝑌) → ∃𝑤𝑋𝑥𝑋 (𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥))))
98ad2antll 726 . . . 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 2737 . . . . . . . . . . . . . 14 (2nd𝑅) = (2nd𝑅)
12 crnghomfo.2 . . . . . . . . . . . . . 14 𝑋 = ran 𝐺
1310, 11, 12crngocom 36219 . . . . . . . . . . . . 13 ((𝑅 ∈ CRingOps ∧ 𝑤𝑋𝑥𝑋) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
14133expb 1119 . . . . . . . . . . . 12 ((𝑅 ∈ CRingOps ∧ (𝑤𝑋𝑥𝑋)) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
15143ad2antl1 1184 . . . . . . . . . . 11 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
1615fveq2d 6813 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = (𝐹‘(𝑥(2nd𝑅)𝑤)))
17 crngorngo 36218 . . . . . . . . . . 11 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18 eqid 2737 . . . . . . . . . . . 12 (2nd𝑆) = (2nd𝑆)
1910, 12, 11, 18rngohommul 36188 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2017, 19syl3anl1 1411 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2110, 12, 11, 18rngohommul 36188 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥𝑋𝑤𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2221ancom2s 647 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2317, 22syl3anl1 1411 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2416, 20, 233eqtr3d 2785 . . . . . . . . 9 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
25 oveq12 7322 . . . . . . . . . 10 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
26 oveq12 7322 . . . . . . . . . . 11 ((𝑧 = (𝐹𝑥) ∧ 𝑦 = (𝐹𝑤)) → (𝑧(2nd𝑆)𝑦) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2726ancoms 459 . . . . . . . . . 10 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑧(2nd𝑆)𝑦) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2825, 27eqeq12d 2753 . . . . . . . . 9 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → ((𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦) ↔ ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤))))
2924, 28syl5ibrcom 246 . . . . . . . 8 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
3029ex 413 . . . . . . 7 ((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
31303expa 1117 . . . . . 6 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3231adantrr 714 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3332rexlimdvv 3201 . . . 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 3192 . 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 36215 . 2 (𝑆 ∈ CRingOps ↔ (𝑆 ∈ RingOps ∧ ∀𝑦𝑌𝑧𝑌 (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦)))
391, 35, 38sylanbrc 583 1 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3062  wrex 3071  ran crn 5606  ontowfo 6461  cfv 6463  (class class class)co 7313  1st c1st 7872  2nd c2nd 7873  RingOpscrngo 36112   RngHom crnghom 36178  CRingOpsccring 36211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fo 6469  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-1st 7874  df-2nd 7875  df-map 8663  df-rngo 36113  df-rngohom 36181  df-com2 36208  df-crngo 36212
This theorem is referenced by: (None)
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