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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 2821 | . 2 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2821 | . 2 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2821 | . 2 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
4 | simp3 1134 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 ∈ 𝐴) | |
5 | ringidss.g | . . . . 5 ⊢ 𝑀 = ((mulGrp‘𝑅) ↾s 𝐴) | |
6 | eqid 2821 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | ringidss.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 6, 7 | mgpbas 19244 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
9 | 5, 8 | ressbas2 16554 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑀)) |
10 | 9 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 = (Base‘𝑀)) |
11 | 4, 10 | eleqtrd 2915 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 ∈ (Base‘𝑀)) |
12 | simp2 1133 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 ⊆ 𝐵) | |
13 | 10, 12 | eqsstrrd 4005 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → (Base‘𝑀) ⊆ 𝐵) |
14 | 13 | sselda 3966 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → 𝑦 ∈ 𝐵) |
15 | fvex 6682 | . . . . . . . 8 ⊢ (Base‘𝑀) ∈ V | |
16 | 10, 15 | eqeltrdi 2921 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 𝐴 ∈ V) |
17 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
18 | 6, 17 | mgpplusg 19242 | . . . . . . . 8 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
19 | 5, 18 | ressplusg 16611 | . . . . . . 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 483 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (.r‘𝑅) = (+g‘𝑀)) |
22 | 21 | oveqd 7172 | . . . 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 19320 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = 𝑦) |
25 | 24 | 3ad2antl1 1181 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑦) = 𝑦) |
26 | 22, 25 | eqtr3d 2858 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ( 1 (+g‘𝑀)𝑦) = 𝑦) |
27 | 14, 26 | syldan 593 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → ( 1 (+g‘𝑀)𝑦) = 𝑦) |
28 | 21 | oveqd 7172 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = (𝑦(+g‘𝑀) 1 )) |
29 | 7, 17, 23 | ringridm 19321 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = 𝑦) |
30 | 29 | 3ad2antl1 1181 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(.r‘𝑅) 1 ) = 𝑦) |
31 | 28, 30 | eqtr3d 2858 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑦(+g‘𝑀) 1 ) = 𝑦) |
32 | 14, 31 | syldan 593 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀) 1 ) = 𝑦) |
33 | 1, 2, 3, 11, 27, 32 | ismgmid2 17877 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 ↾s cress 16483 +gcplusg 16564 .rcmulr 16565 0gc0g 16712 mulGrpcmgp 19238 1rcur 19250 Ringcrg 19296 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mgp 19239 df-ur 19251 df-ring 19298 |
This theorem is referenced by: unitgrpid 19418 cnmgpid 20606 xrge0iifmhm 31182 |
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