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Mirrors > Home > ILE Home > Th. List > ress0g | GIF version |
Description: 0g is unaffected by restriction. This is a bit more generic than submnd0 12799. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
Ref | Expression |
---|---|
ress0g.s | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
ress0g.b | ⊢ 𝐵 = (Base‘𝑅) |
ress0g.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
ress0g | ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 = (0g‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ress0g.s | . . . 4 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
2 | 1 | a1i 9 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑆 = (𝑅 ↾s 𝐴)) |
3 | ress0g.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐵 = (Base‘𝑅)) |
5 | simp1 997 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑅 ∈ Mnd) | |
6 | simp3 999 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 2, 4, 5, 6 | ressbas2d 12522 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘𝑆)) |
8 | eqidd 2178 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (+g‘𝑅) = (+g‘𝑅)) | |
9 | basfn 12514 | . . . . . 6 ⊢ Base Fn V | |
10 | 5 | elexd 2750 | . . . . . 6 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑅 ∈ V) |
11 | funfvex 5532 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V) | |
12 | 11 | funfni 5316 | . . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V) |
13 | 9, 10, 12 | sylancr 414 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (Base‘𝑅) ∈ V) |
14 | 3, 13 | eqeltrid 2264 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ V) |
15 | 14, 6 | ssexd 4143 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
16 | 2, 8, 15, 5 | ressplusgd 12581 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (+g‘𝑅) = (+g‘𝑆)) |
17 | simp2 998 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 ∈ 𝐴) | |
18 | simpl1 1000 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ Mnd) | |
19 | 6 | sselda 3155 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
20 | eqid 2177 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
21 | ress0g.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
22 | 3, 20, 21 | mndlid 12790 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
23 | 18, 19, 22 | syl2anc 411 | . 2 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
24 | 3, 20, 21 | mndrid 12791 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑅) 0 ) = 𝑥) |
25 | 18, 19, 24 | syl2anc 411 | . 2 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(+g‘𝑅) 0 ) = 𝑥) |
26 | 7, 16, 17, 23, 25 | grpidd 12756 | 1 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 = (0g‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 Fn wfn 5211 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 ↾s cress 12457 +gcplusg 12530 0gc0g 12695 Mndcmnd 12771 |
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-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-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-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-iota 5178 df-fun 5218 df-fn 5219 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-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-iress 12464 df-plusg 12543 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 |
This theorem is referenced by: submnd0 12799 zring0 13381 |
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