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Mirrors > Home > MPE Home > Th. List > subrg1 | Structured version Visualization version GIF version |
Description: A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrg1.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
subrg1.2 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
subrg1 | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2795 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | 1 | subrg1cl 19238 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐴) |
3 | subrg1.1 | . . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
4 | 3 | subrgbas 19239 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
5 | 2, 4 | eleqtrd 2885 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ (Base‘𝑆)) |
6 | eqid 2795 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 19231 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
8 | 4, 7 | eqsstrrd 3931 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅)) |
9 | 8 | sselda 3893 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅)) |
10 | subrgrcl 19235 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
11 | eqid 2795 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | 6, 11, 1 | ringidmlem 19015 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥)) |
13 | 10, 12 | sylan 580 | . . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥)) |
14 | 3, 11 | ressmulr 16459 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
15 | 14 | oveqd 7038 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((1r‘𝑅)(.r‘𝑅)𝑥) = ((1r‘𝑅)(.r‘𝑆)𝑥)) |
16 | 15 | eqeq1d 2797 | . . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ↔ ((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥)) |
17 | 14 | oveqd 7038 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)(1r‘𝑅)) = (𝑥(.r‘𝑆)(1r‘𝑅))) |
18 | 17 | eqeq1d 2797 | . . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥 ↔ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) |
19 | 16, 18 | anbi12d 630 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥) ↔ (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥))) |
20 | 19 | biimpa 477 | . . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥)) → (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) |
21 | 13, 20 | syldan 591 | . . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) |
22 | 9, 21 | syldan 591 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Base‘𝑆)) → (((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) |
23 | 22 | ralrimiva 3149 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ (Base‘𝑆)(((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) |
24 | 3 | subrgring 19233 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) |
25 | eqid 2795 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
26 | eqid 2795 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
27 | eqid 2795 | . . . . 5 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
28 | 25, 26, 27 | isringid 19018 | . . . 4 ⊢ (𝑆 ∈ Ring → (((1r‘𝑅) ∈ (Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)(((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) ↔ (1r‘𝑆) = (1r‘𝑅))) |
29 | 24, 28 | syl 17 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (((1r‘𝑅) ∈ (Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)(((1r‘𝑅)(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(1r‘𝑅)) = 𝑥)) ↔ (1r‘𝑆) = (1r‘𝑅))) |
30 | 5, 23, 29 | mpbi2and 708 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑆) = (1r‘𝑅)) |
31 | subrg1.2 | . 2 ⊢ 1 = (1r‘𝑅) | |
32 | 30, 31 | syl6reqr 2850 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 ‘cfv 6230 (class class class)co 7021 Basecbs 16317 ↾s cress 16318 .rcmulr 16400 1rcur 18946 Ringcrg 18992 SubRingcsubrg 19226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-0g 16549 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-subg 18035 df-mgp 18935 df-ur 18947 df-ring 18994 df-subrg 19228 |
This theorem is referenced by: subrguss 19245 subrginv 19246 subrgunit 19248 subsubrg 19256 sralmod 19654 subrgnzr 19735 ressascl 19817 mpl1 19917 subrgmvr 19934 gzrngunitlem 20297 zring1 20315 re1r 20444 scmatsrng1 20821 scmatmhm 20832 clm1 23365 isclmp 23389 qrng1 25885 subrgchr 30524 |
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