Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crngohomfo Structured version   Visualization version   GIF version

Theorem crngohomfo 36465
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 767 . 2 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ RingOps)
2 foelrn 7056 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑦𝑌) → ∃𝑤𝑋 𝑦 = (𝐹𝑤))
32ex 413 . . . . . . 7 (𝐹:𝑋onto𝑌 → (𝑦𝑌 → ∃𝑤𝑋 𝑦 = (𝐹𝑤)))
4 foelrn 7056 . . . . . . . 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 3217 . . . . . 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 727 . . . 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 2736 . . . . . . . . . . . . . 14 (2nd𝑅) = (2nd𝑅)
12 crnghomfo.2 . . . . . . . . . . . . . 14 𝑋 = ran 𝐺
1310, 11, 12crngocom 36460 . . . . . . . . . . . . 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 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝑤(2nd𝑅)𝑥) = (𝑥(2nd𝑅)𝑤))
1615fveq2d 6846 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = (𝐹‘(𝑥(2nd𝑅)𝑤)))
17 crngorngo 36459 . . . . . . . . . . 11 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
18 eqid 2736 . . . . . . . . . . . 12 (2nd𝑆) = (2nd𝑆)
1910, 12, 11, 18rngohommul 36429 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2017, 19syl3anl1 1412 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑤(2nd𝑅)𝑥)) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
2110, 12, 11, 18rngohommul 36429 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥𝑋𝑤𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2221ancom2s 648 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2317, 22syl3anl1 1412 . . . . . . . . . 10 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → (𝐹‘(𝑥(2nd𝑅)𝑤)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
2416, 20, 233eqtr3d 2784 . . . . . . . . 9 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑤𝑋𝑥𝑋)) → ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑤)))
25 oveq12 7366 . . . . . . . . . 10 ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = ((𝐹𝑤)(2nd𝑆)(𝐹𝑥)))
26 oveq12 7366 . . . . . . . . . . 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 2752 . . . . . . . . 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 1118 . . . . . 6 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3231adantrr 715 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → ((𝑤𝑋𝑥𝑋) → ((𝑦 = (𝐹𝑤) ∧ 𝑧 = (𝐹𝑥)) → (𝑦(2nd𝑆)𝑧) = (𝑧(2nd𝑆)𝑦))))
3332rexlimdvv 3204 . . . 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 3195 . 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 36456 . 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 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  ran crn 5634  ontowfo 6494  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  RingOpscrngo 36353   RngHom crnghom 36419  CRingOpsccring 36452
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-rngo 36354  df-rngohom 36422  df-com2 36449  df-crngo 36453
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator