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Mirrors > Home > MPE Home > Th. List > subrgsubm | Structured version Visualization version GIF version |
Description: A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
subrgsubm.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
subrgsubm | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | 1 | subrgss 19465 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
3 | eqid 2818 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 3 | subrg1cl 19472 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐴) |
5 | subrgrcl 19469 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
6 | eqid 2818 | . . . . 5 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | subrgsubm.1 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | 6, 7 | mgpress 19179 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑀 ↾s 𝐴) = (mulGrp‘(𝑅 ↾s 𝐴))) |
9 | 5, 8 | mpancom 684 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑀 ↾s 𝐴) = (mulGrp‘(𝑅 ↾s 𝐴))) |
10 | 6 | subrgring 19467 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) |
11 | eqid 2818 | . . . . 5 ⊢ (mulGrp‘(𝑅 ↾s 𝐴)) = (mulGrp‘(𝑅 ↾s 𝐴)) | |
12 | 11 | ringmgp 19232 | . . . 4 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (mulGrp‘(𝑅 ↾s 𝐴)) ∈ Mnd) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (mulGrp‘(𝑅 ↾s 𝐴)) ∈ Mnd) |
14 | 9, 13 | eqeltrd 2910 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑀 ↾s 𝐴) ∈ Mnd) |
15 | 7 | ringmgp 19232 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
16 | 7, 1 | mgpbas 19174 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
17 | 7, 3 | ringidval 19182 | . . . 4 ⊢ (1r‘𝑅) = (0g‘𝑀) |
18 | eqid 2818 | . . . 4 ⊢ (𝑀 ↾s 𝐴) = (𝑀 ↾s 𝐴) | |
19 | 16, 17, 18 | issubm2 17957 | . . 3 ⊢ (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd))) |
20 | 5, 15, 19 | 3syl 18 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴 ∧ (𝑀 ↾s 𝐴) ∈ Mnd))) |
21 | 2, 4, 14, 20 | mpbir3and 1334 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 Mndcmnd 17899 SubMndcsubmnd 17943 mulGrpcmgp 19168 1rcur 19180 Ringcrg 19226 SubRingcsubrg 19460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mgp 19169 df-ur 19181 df-ring 19228 df-subrg 19462 |
This theorem is referenced by: resrhm 19493 rhmima 19495 mplbas2 20179 zrhpsgnmhm 20656 m2cpmmhm 21281 cmodscexp 23652 plypf1 24729 wilthlem2 25573 wilthlem3 25574 lgsqrlem1 25849 lgseisenlem4 25881 dchrisum0flblem1 26011 |
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