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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 2735 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | 1 | subrgss 20589 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
3 | eqid 2735 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 3 | subrg1cl 20597 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐴) |
5 | subrgrcl 20593 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
6 | eqid 2735 | . . . . 5 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | subrgsubm.1 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
8 | 6, 7 | mgpress 20167 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑀 ↾s 𝐴) = (mulGrp‘(𝑅 ↾s 𝐴))) |
9 | 5, 8 | mpancom 688 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑀 ↾s 𝐴) = (mulGrp‘(𝑅 ↾s 𝐴))) |
10 | 6 | subrgring 20591 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) |
11 | eqid 2735 | . . . . 5 ⊢ (mulGrp‘(𝑅 ↾s 𝐴)) = (mulGrp‘(𝑅 ↾s 𝐴)) | |
12 | 11 | ringmgp 20257 | . . . 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 2839 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑀 ↾s 𝐴) ∈ Mnd) |
15 | 7 | ringmgp 20257 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd) |
16 | 7, 1 | mgpbas 20158 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑀) |
17 | 7, 3 | ringidval 20201 | . . . 4 ⊢ (1r‘𝑅) = (0g‘𝑀) |
18 | eqid 2735 | . . . 4 ⊢ (𝑀 ↾s 𝐴) = (𝑀 ↾s 𝐴) | |
19 | 16, 17, 18 | issubm2 18830 | . . 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 1341 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 Mndcmnd 18760 SubMndcsubmnd 18808 mulGrpcmgp 20152 1rcur 20199 Ringcrg 20251 SubRingcsubrg 20586 |
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-submnd 18810 df-mgp 20153 df-ur 20200 df-ring 20253 df-subrg 20587 |
This theorem is referenced by: resrhm 20618 resrhm2b 20619 rhmima 20621 zrhpsgnmhm 21620 mplbas2 22078 m2cpmmhm 22767 cmodscexp 25168 plypf1 26266 wilthlem2 27127 wilthlem3 27128 lgsqrlem1 27405 lgseisenlem4 27437 dchrisum0flblem1 27567 elrgspnlem4 33235 |
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