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Theorem rncrhmcl 40632
Description: The range of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.)
Hypotheses
Ref Expression
rncrhmcl.c 𝐶 = (𝑁s ran 𝐹)
rncrhmcl.h (𝜑𝐹 ∈ (𝑀 RingHom 𝑁))
rncrhmcl.m (𝜑𝑀 ∈ CRing)
Assertion
Ref Expression
rncrhmcl (𝜑𝐶 ∈ CRing)

Proof of Theorem rncrhmcl
Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rncrhmcl.h . . 3 (𝜑𝐹 ∈ (𝑀 RingHom 𝑁))
2 rnrhmsubrg 20207 . . 3 (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
3 rncrhmcl.c . . . 4 𝐶 = (𝑁s ran 𝐹)
43subrgring 20178 . . 3 (ran 𝐹 ∈ (SubRing‘𝑁) → 𝐶 ∈ Ring)
51, 2, 43syl 18 . 2 (𝜑𝐶 ∈ Ring)
63ressbasss2 40600 . . . . . 6 (Base‘𝐶) ⊆ ran 𝐹
76sseli 3939 . . . . 5 (𝑥 ∈ (Base‘𝐶) → 𝑥 ∈ ran 𝐹)
86sseli 3939 . . . . 5 (𝑦 ∈ (Base‘𝐶) → 𝑦 ∈ ran 𝐹)
97, 8anim12i 614 . . . 4 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹))
10 eqid 2738 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
11 eqid 2738 . . . . . . . . 9 (Base‘𝑁) = (Base‘𝑁)
1210, 11rhmf 20111 . . . . . . . 8 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
131, 12syl 17 . . . . . . 7 (𝜑𝐹:(Base‘𝑀)⟶(Base‘𝑁))
1413ffnd 6667 . . . . . 6 (𝜑𝐹 Fn (Base‘𝑀))
15 simpl 484 . . . . . 6 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹) → 𝑥 ∈ ran 𝐹)
16 fvelrnb 6901 . . . . . . 7 (𝐹 Fn (Base‘𝑀) → (𝑥 ∈ ran 𝐹 ↔ ∃𝑎 ∈ (Base‘𝑀)(𝐹𝑎) = 𝑥))
1716biimpa 478 . . . . . 6 ((𝐹 Fn (Base‘𝑀) ∧ 𝑥 ∈ ran 𝐹) → ∃𝑎 ∈ (Base‘𝑀)(𝐹𝑎) = 𝑥)
1814, 15, 17syl2an 597 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → ∃𝑎 ∈ (Base‘𝑀)(𝐹𝑎) = 𝑥)
19 simpr 486 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
20 fvelrnb 6901 . . . . . . . . 9 (𝐹 Fn (Base‘𝑀) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑏 ∈ (Base‘𝑀)(𝐹𝑏) = 𝑦))
2120biimpa 478 . . . . . . . 8 ((𝐹 Fn (Base‘𝑀) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑏 ∈ (Base‘𝑀)(𝐹𝑏) = 𝑦)
2214, 19, 21syl2an 597 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → ∃𝑏 ∈ (Base‘𝑀)(𝐹𝑏) = 𝑦)
2322adantr 482 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) → ∃𝑏 ∈ (Base‘𝑀)(𝐹𝑏) = 𝑦)
24 rncrhmcl.m . . . . . . . . . . 11 (𝜑𝑀 ∈ CRing)
2524ad3antrrr 729 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → 𝑀 ∈ CRing)
26 simplrl 776 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → 𝑎 ∈ (Base‘𝑀))
27 simprl 770 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → 𝑏 ∈ (Base‘𝑀))
28 eqid 2738 . . . . . . . . . . 11 (.r𝑀) = (.r𝑀)
2910, 28crngcom 19936 . . . . . . . . . 10 ((𝑀 ∈ CRing ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(.r𝑀)𝑏) = (𝑏(.r𝑀)𝑎))
3025, 26, 27, 29syl3anc 1372 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (𝑎(.r𝑀)𝑏) = (𝑏(.r𝑀)𝑎))
3130fveq2d 6844 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (𝐹‘(𝑎(.r𝑀)𝑏)) = (𝐹‘(𝑏(.r𝑀)𝑎)))
321ad3antrrr 729 . . . . . . . . 9 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → 𝐹 ∈ (𝑀 RingHom 𝑁))
33 eqid 2738 . . . . . . . . . 10 (.r𝑁) = (.r𝑁)
3410, 28, 33rhmmul 20112 . . . . . . . . 9 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝐹‘(𝑎(.r𝑀)𝑏)) = ((𝐹𝑎)(.r𝑁)(𝐹𝑏)))
3532, 26, 27, 34syl3anc 1372 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (𝐹‘(𝑎(.r𝑀)𝑏)) = ((𝐹𝑎)(.r𝑁)(𝐹𝑏)))
3610, 28, 33rhmmul 20112 . . . . . . . . 9 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑏 ∈ (Base‘𝑀) ∧ 𝑎 ∈ (Base‘𝑀)) → (𝐹‘(𝑏(.r𝑀)𝑎)) = ((𝐹𝑏)(.r𝑁)(𝐹𝑎)))
3732, 27, 26, 36syl3anc 1372 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (𝐹‘(𝑏(.r𝑀)𝑎)) = ((𝐹𝑏)(.r𝑁)(𝐹𝑎)))
3831, 35, 373eqtr3d 2786 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → ((𝐹𝑎)(.r𝑁)(𝐹𝑏)) = ((𝐹𝑏)(.r𝑁)(𝐹𝑎)))
39 rnexg 7834 . . . . . . . . . 10 (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ V)
403, 33ressmulr 17148 . . . . . . . . . 10 (ran 𝐹 ∈ V → (.r𝑁) = (.r𝐶))
411, 39, 403syl 18 . . . . . . . . 9 (𝜑 → (.r𝑁) = (.r𝐶))
4241ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (.r𝑁) = (.r𝐶))
43 simplrr 777 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (𝐹𝑎) = 𝑥)
44 simprr 772 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (𝐹𝑏) = 𝑦)
4542, 43, 44oveq123d 7373 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → ((𝐹𝑎)(.r𝑁)(𝐹𝑏)) = (𝑥(.r𝐶)𝑦))
4642, 44, 43oveq123d 7373 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → ((𝐹𝑏)(.r𝑁)(𝐹𝑎)) = (𝑦(.r𝐶)𝑥))
4738, 45, 463eqtr3d 2786 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) ∧ (𝑏 ∈ (Base‘𝑀) ∧ (𝐹𝑏) = 𝑦)) → (𝑥(.r𝐶)𝑦) = (𝑦(.r𝐶)𝑥))
4823, 47rexlimddv 3157 . . . . 5 (((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) ∧ (𝑎 ∈ (Base‘𝑀) ∧ (𝐹𝑎) = 𝑥)) → (𝑥(.r𝐶)𝑦) = (𝑦(.r𝐶)𝑥))
4918, 48rexlimddv 3157 . . . 4 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → (𝑥(.r𝐶)𝑦) = (𝑦(.r𝐶)𝑥))
509, 49sylan2 594 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(.r𝐶)𝑦) = (𝑦(.r𝐶)𝑥))
5150ralrimivva 3196 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r𝐶)𝑦) = (𝑦(.r𝐶)𝑥))
52 eqid 2738 . . 3 (Base‘𝐶) = (Base‘𝐶)
53 eqid 2738 . . 3 (.r𝐶) = (.r𝐶)
5452, 53iscrng2 19937 . 2 (𝐶 ∈ CRing ↔ (𝐶 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r𝐶)𝑦) = (𝑦(.r𝐶)𝑥)))
555, 51, 54sylanbrc 584 1 (𝜑𝐶 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3063  wrex 3072  Vcvv 3444  ran crn 5633   Fn wfn 6489  wf 6490  cfv 6494  (class class class)co 7352  Basecbs 17043  s cress 17072  .rcmulr 17094  Ringcrg 19918  CRingccrg 19919   RingHom crh 20096  SubRingcsubrg 20171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-cnex 11066  ax-resscn 11067  ax-1cn 11068  ax-icn 11069  ax-addcl 11070  ax-addrcl 11071  ax-mulcl 11072  ax-mulrcl 11073  ax-mulcom 11074  ax-addass 11075  ax-mulass 11076  ax-distr 11077  ax-i2m1 11078  ax-1ne0 11079  ax-1rid 11080  ax-rnegex 11081  ax-rrecex 11082  ax-cnre 11083  ax-pre-lttri 11084  ax-pre-lttrn 11085  ax-pre-ltadd 11086  ax-pre-mulgt0 11087
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7796  df-2nd 7915  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349  df-er 8607  df-map 8726  df-en 8843  df-dom 8844  df-sdom 8845  df-pnf 11150  df-mnf 11151  df-xr 11152  df-ltxr 11153  df-le 11154  df-sub 11346  df-neg 11347  df-nn 12113  df-2 12175  df-3 12176  df-sets 16996  df-slot 17014  df-ndx 17026  df-base 17044  df-ress 17073  df-plusg 17106  df-mulr 17107  df-0g 17283  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-mhm 18561  df-submnd 18562  df-grp 18711  df-minusg 18712  df-subg 18884  df-ghm 18965  df-cmn 19523  df-mgp 19856  df-ur 19873  df-ring 19920  df-cring 19921  df-rnghom 20099  df-subrg 20173
This theorem is referenced by:  riccrng1  40638
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