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Theorem subrdom 33269
Description: A subring of a domain is a domain. (Contributed by Thierry Arnoux, 18-May-2025.)
Hypotheses
Ref Expression
subrdom.1 (𝜑𝑅 ∈ Domn)
subrdom.2 (𝜑𝑆 ∈ (SubRing‘𝑅))
Assertion
Ref Expression
subrdom (𝜑 → (𝑅s 𝑆) ∈ Domn)

Proof of Theorem subrdom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrdom.1 . . . 4 (𝜑𝑅 ∈ Domn)
2 domnnzr 20723 . . . 4 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
31, 2syl 17 . . 3 (𝜑𝑅 ∈ NzRing)
4 subrdom.2 . . 3 (𝜑𝑆 ∈ (SubRing‘𝑅))
5 eqid 2735 . . . 4 (𝑅s 𝑆) = (𝑅s 𝑆)
65subrgnzr 20611 . . 3 ((𝑅 ∈ NzRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → (𝑅s 𝑆) ∈ NzRing)
73, 4, 6syl2anc 584 . 2 (𝜑 → (𝑅s 𝑆) ∈ NzRing)
81ad3antrrr 730 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑅 ∈ Domn)
9 eqid 2735 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
109subrgss 20589 . . . . . . . . . 10 (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ (Base‘𝑅))
114, 10syl 17 . . . . . . . . 9 (𝜑𝑆 ⊆ (Base‘𝑅))
1211ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑆 ⊆ (Base‘𝑅))
13 simpllr 776 . . . . . . . . 9 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑥 ∈ (Base‘(𝑅s 𝑆)))
145, 9ressbas2 17283 . . . . . . . . . . 11 (𝑆 ⊆ (Base‘𝑅) → 𝑆 = (Base‘(𝑅s 𝑆)))
1511, 14syl 17 . . . . . . . . . 10 (𝜑𝑆 = (Base‘(𝑅s 𝑆)))
1615ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑆 = (Base‘(𝑅s 𝑆)))
1713, 16eleqtrrd 2842 . . . . . . . 8 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑥𝑆)
1812, 17sseldd 3996 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑥 ∈ (Base‘𝑅))
19 simplr 769 . . . . . . . . 9 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑦 ∈ (Base‘(𝑅s 𝑆)))
2019, 16eleqtrrd 2842 . . . . . . . 8 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑦𝑆)
2112, 20sseldd 3996 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → 𝑦 ∈ (Base‘𝑅))
22 simpr 484 . . . . . . . 8 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆)))
234elexd 3502 . . . . . . . . . . 11 (𝜑𝑆 ∈ V)
24 eqid 2735 . . . . . . . . . . . 12 (.r𝑅) = (.r𝑅)
255, 24ressmulr 17353 . . . . . . . . . . 11 (𝑆 ∈ V → (.r𝑅) = (.r‘(𝑅s 𝑆)))
2623, 25syl 17 . . . . . . . . . 10 (𝜑 → (.r𝑅) = (.r‘(𝑅s 𝑆)))
2726oveqd 7448 . . . . . . . . 9 (𝜑 → (𝑥(.r𝑅)𝑦) = (𝑥(.r‘(𝑅s 𝑆))𝑦))
2827ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (𝑥(.r𝑅)𝑦) = (𝑥(.r‘(𝑅s 𝑆))𝑦))
29 subrgrcl 20593 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
30 ringmnd 20261 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
314, 29, 303syl 18 . . . . . . . . . 10 (𝜑𝑅 ∈ Mnd)
32 subrgsubg 20594 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ∈ (SubGrp‘𝑅))
33 eqid 2735 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
3433subg0cl 19165 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝑅) → (0g𝑅) ∈ 𝑆)
354, 32, 343syl 18 . . . . . . . . . 10 (𝜑 → (0g𝑅) ∈ 𝑆)
365, 9, 33ress0g 18788 . . . . . . . . . 10 ((𝑅 ∈ Mnd ∧ (0g𝑅) ∈ 𝑆𝑆 ⊆ (Base‘𝑅)) → (0g𝑅) = (0g‘(𝑅s 𝑆)))
3731, 35, 11, 36syl3anc 1370 . . . . . . . . 9 (𝜑 → (0g𝑅) = (0g‘(𝑅s 𝑆)))
3837ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (0g𝑅) = (0g‘(𝑅s 𝑆)))
3922, 28, 383eqtr4d 2785 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
409, 24, 33domneq0 20725 . . . . . . . 8 ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) = (0g𝑅) ↔ (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅))))
4140biimpa 476 . . . . . . 7 (((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) = (0g𝑅)) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))
428, 18, 21, 39, 41syl31anc 1372 . . . . . 6 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))
4338eqeq2d 2746 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (𝑥 = (0g𝑅) ↔ 𝑥 = (0g‘(𝑅s 𝑆))))
4438eqeq2d 2746 . . . . . . 7 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (𝑦 = (0g𝑅) ↔ 𝑦 = (0g‘(𝑅s 𝑆))))
4543, 44orbi12d 918 . . . . . 6 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → ((𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)) ↔ (𝑥 = (0g‘(𝑅s 𝑆)) ∨ 𝑦 = (0g‘(𝑅s 𝑆)))))
4642, 45mpbid 232 . . . . 5 ((((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) ∧ (𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆))) → (𝑥 = (0g‘(𝑅s 𝑆)) ∨ 𝑦 = (0g‘(𝑅s 𝑆))))
4746ex 412 . . . 4 (((𝜑𝑥 ∈ (Base‘(𝑅s 𝑆))) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆))) → ((𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆)) → (𝑥 = (0g‘(𝑅s 𝑆)) ∨ 𝑦 = (0g‘(𝑅s 𝑆)))))
4847anasss 466 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝑅s 𝑆)) ∧ 𝑦 ∈ (Base‘(𝑅s 𝑆)))) → ((𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆)) → (𝑥 = (0g‘(𝑅s 𝑆)) ∨ 𝑦 = (0g‘(𝑅s 𝑆)))))
4948ralrimivva 3200 . 2 (𝜑 → ∀𝑥 ∈ (Base‘(𝑅s 𝑆))∀𝑦 ∈ (Base‘(𝑅s 𝑆))((𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆)) → (𝑥 = (0g‘(𝑅s 𝑆)) ∨ 𝑦 = (0g‘(𝑅s 𝑆)))))
50 eqid 2735 . . 3 (Base‘(𝑅s 𝑆)) = (Base‘(𝑅s 𝑆))
51 eqid 2735 . . 3 (.r‘(𝑅s 𝑆)) = (.r‘(𝑅s 𝑆))
52 eqid 2735 . . 3 (0g‘(𝑅s 𝑆)) = (0g‘(𝑅s 𝑆))
5350, 51, 52isdomn 20722 . 2 ((𝑅s 𝑆) ∈ Domn ↔ ((𝑅s 𝑆) ∈ NzRing ∧ ∀𝑥 ∈ (Base‘(𝑅s 𝑆))∀𝑦 ∈ (Base‘(𝑅s 𝑆))((𝑥(.r‘(𝑅s 𝑆))𝑦) = (0g‘(𝑅s 𝑆)) → (𝑥 = (0g‘(𝑅s 𝑆)) ∨ 𝑦 = (0g‘(𝑅s 𝑆))))))
547, 49, 53sylanbrc 583 1 (𝜑 → (𝑅s 𝑆) ∈ Domn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  .rcmulr 17299  0gc0g 17486  Mndcmnd 18760  SubGrpcsubg 19151  Ringcrg 20251  NzRingcnzr 20529  SubRingcsubrg 20586  Domncdomn 20709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-nzr 20530  df-subrg 20587  df-domn 20712
This theorem is referenced by:  subridom  33270
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