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Mirrors > Home > MPE Home > Th. List > ringidss | Structured version Visualization version GIF version |
Description: A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
ringidss.g | ⊢ 𝑀 = ((mulGrp‘𝑅) ↾s 𝐴) |
ringidss.b | ⊢ 𝐵 = (Base‘𝑅) |
ringidss.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ringidss | ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 = (0g‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . 2 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2736 | . 2 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2736 | . 2 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
4 | simp3 1138 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 ∈ 𝐴) | |
5 | ringidss.g | . . . . 5 ⊢ 𝑀 = ((mulGrp‘𝑅) ↾s 𝐴) | |
6 | eqid 2736 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | ringidss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 6, 7 | mgpbas 19893 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
9 | 5, 8 | ressbas2 17112 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑀)) |
10 | 9 | 3ad2ant2 1134 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 = (Base‘𝑀)) |
11 | 4, 10 | eleqtrd 2840 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 ∈ (Base‘𝑀)) |
12 | simp2 1137 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 ⊆ 𝐵) | |
13 | 10, 12 | eqsstrrd 3981 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (Base‘𝑀) ⊆ 𝐵) |
14 | 13 | sselda 3942 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → 𝑦 ∈ 𝐵) |
15 | fvex 6852 | . . . . . . . 8 ⊢ (Base‘𝑀) ∈ V | |
16 | 10, 15 | eqeltrdi 2846 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 ∈ V) |
17 | eqid 2736 | . . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
18 | 6, 17 | mgpplusg 19891 | . . . . . . . 8 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
19 | 5, 18 | ressplusg 17163 | . . . . . . 7 ⊢ (𝐴 ∈ V → (.r‘𝑅) = (+g‘𝑀)) |
20 | 16, 19 | syl 17 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (.r‘𝑅) = (+g‘𝑀)) |
21 | 20 | adantr 481 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (.r‘𝑅) = (+g‘𝑀)) |
22 | 21 | oveqd 7370 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = ( 1 (+g‘𝑀)𝑦)) |
23 | ringidss.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
24 | 7, 17, 23 | ringlidm 19978 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = 𝑦) |
25 | 24 | 3ad2antl1 1185 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = 𝑦) |
26 | 22, 25 | eqtr3d 2778 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (+g‘𝑀)𝑦) = 𝑦) |
27 | 14, 26 | syldan 591 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → ( 1 (+g‘𝑀)𝑦) = 𝑦) |
28 | 21 | oveqd 7370 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = (𝑦(+g‘𝑀) 1 )) |
29 | 7, 17, 23 | ringridm 19979 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = 𝑦) |
30 | 29 | 3ad2antl1 1185 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = 𝑦) |
31 | 28, 30 | eqtr3d 2778 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(+g‘𝑀) 1 ) = 𝑦) |
32 | 14, 31 | syldan 591 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀) 1 ) = 𝑦) |
33 | 1, 2, 3, 11, 27, 32 | ismgmid2 18515 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 ↾s cress 17104 +gcplusg 17125 .rcmulr 17126 0gc0g 17313 mulGrpcmgp 19887 1rcur 19904 Ringcrg 19950 |
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 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-0g 17315 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mgp 19888 df-ur 19905 df-ring 19952 |
This theorem is referenced by: unitgrpid 20083 cnmgpid 20844 xrge0iifmhm 32389 |
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