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Mirrors > Home > ILE Home > Th. List > ringressid | GIF version |
Description: A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12524. (Contributed by Jim Kingdon, 28-Feb-2025.) |
Ref | Expression |
---|---|
ringressid.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
ringressid | ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2178 | . . 3 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵)) | |
2 | ringressid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝐺 ∈ Ring → 𝐵 = (Base‘𝐺)) |
4 | id 19 | . . 3 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Ring) | |
5 | ssidd 3176 | . . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ⊆ 𝐵) | |
6 | 1, 3, 4, 5 | ressbas2d 12522 | . 2 ⊢ (𝐺 ∈ Ring → 𝐵 = (Base‘(𝐺 ↾s 𝐵))) |
7 | eqidd 2178 | . . 3 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘𝐺)) | |
8 | basfn 12514 | . . . . 5 ⊢ Base Fn V | |
9 | elex 2748 | . . . . 5 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ V) | |
10 | funfvex 5532 | . . . . . 6 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V) | |
11 | 10 | funfni 5316 | . . . . 5 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V) |
12 | 8, 9, 11 | sylancr 414 | . . . 4 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) ∈ V) |
13 | 2, 12 | eqeltrid 2264 | . . 3 ⊢ (𝐺 ∈ Ring → 𝐵 ∈ V) |
14 | 1, 7, 13, 4 | ressplusgd 12581 | . 2 ⊢ (𝐺 ∈ Ring → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐵))) |
15 | eqid 2177 | . . . 4 ⊢ (𝐺 ↾s 𝐵) = (𝐺 ↾s 𝐵) | |
16 | eqid 2177 | . . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
17 | 15, 16 | ressmulrg 12597 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐺 ∈ Ring) → (.r‘𝐺) = (.r‘(𝐺 ↾s 𝐵))) |
18 | 13, 17 | mpancom 422 | . 2 ⊢ (𝐺 ∈ Ring → (.r‘𝐺) = (.r‘(𝐺 ↾s 𝐵))) |
19 | ringgrp 13137 | . . 3 ⊢ (𝐺 ∈ Ring → 𝐺 ∈ Grp) | |
20 | 2 | grpressid 12885 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝐵) ∈ Grp) |
21 | 19, 20 | syl 14 | . 2 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Grp) |
22 | 2, 16 | ringcl 13149 | . 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐺)𝑦) ∈ 𝐵) |
23 | 2, 16 | ringass 13152 | . 2 ⊢ ((𝐺 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(.r‘𝐺)𝑦)(.r‘𝐺)𝑧) = (𝑥(.r‘𝐺)(𝑦(.r‘𝐺)𝑧))) |
24 | eqid 2177 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
25 | 2, 24, 16 | ringdi 13154 | . 2 ⊢ ((𝐺 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥(.r‘𝐺)(𝑦(+g‘𝐺)𝑧)) = ((𝑥(.r‘𝐺)𝑦)(+g‘𝐺)(𝑥(.r‘𝐺)𝑧))) |
26 | 2, 24, 16 | ringdir 13155 | . 2 ⊢ ((𝐺 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(.r‘𝐺)𝑧) = ((𝑥(.r‘𝐺)𝑧)(+g‘𝐺)(𝑦(.r‘𝐺)𝑧))) |
27 | eqid 2177 | . . 3 ⊢ (1r‘𝐺) = (1r‘𝐺) | |
28 | 2, 27 | ringidcl 13156 | . 2 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ 𝐵) |
29 | 2, 16, 27 | ringlidm 13159 | . 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((1r‘𝐺)(.r‘𝐺)𝑥) = 𝑥) |
30 | 2, 16, 27 | ringridm 13160 | . 2 ⊢ ((𝐺 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝐺)(1r‘𝐺)) = 𝑥) |
31 | 6, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30 | isringd 13173 | 1 ⊢ (𝐺 ∈ Ring → (𝐺 ↾s 𝐵) ∈ Ring) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 Fn wfn 5211 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 ↾s cress 12457 +gcplusg 12530 .rcmulr 12531 Grpcgrp 12831 1rcur 13095 Ringcrg 13132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-iress 12464 df-plusg 12543 df-mulr 12544 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-minusg 12835 df-mgp 13084 df-ur 13096 df-ring 13134 |
This theorem is referenced by: subrgid 13304 |
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