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Mirrors > Home > MPE Home > Th. List > ress0g | Structured version Visualization version GIF version |
Description: 0g is unaffected by restriction. This is a bit more generic than submnd0 17932. (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 | ress0g.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 1, 2 | ressbas2 16547 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆)) |
4 | 3 | 3ad2ant3 1132 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘𝑆)) |
5 | simp3 1135 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
6 | 2 | fvexi 6659 | . . . 4 ⊢ 𝐵 ∈ V |
7 | ssexg 5191 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
8 | 5, 6, 7 | sylancl 589 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
9 | eqid 2798 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
10 | 1, 9 | ressplusg 16604 | . . 3 ⊢ (𝐴 ∈ V → (+g‘𝑅) = (+g‘𝑆)) |
11 | 8, 10 | syl 17 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (+g‘𝑅) = (+g‘𝑆)) |
12 | simp2 1134 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 ∈ 𝐴) | |
13 | simpl1 1188 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ Mnd) | |
14 | 5 | sselda 3915 | . . 3 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
15 | ress0g.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
16 | 2, 9, 15 | mndlid 17923 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
17 | 13, 14, 16 | syl2anc 587 | . 2 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → ( 0 (+g‘𝑅)𝑥) = 𝑥) |
18 | 2, 9, 15 | mndrid 17924 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝑅) 0 ) = 𝑥) |
19 | 13, 14, 18 | syl2anc 587 | . 2 ⊢ (((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(+g‘𝑅) 0 ) = 𝑥) |
20 | 4, 11, 12, 17, 19 | grpidd 17873 | 1 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 = (0g‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 0gc0g 16705 Mndcmnd 17903 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 |
This theorem is referenced by: nn0srg 20161 rge0srg 20162 zring0 20173 re0g 20301 ressnm 30664 psgnid 30789 cnmsgn0g 30838 altgnsg 30841 xrge0slmod 30968 drgext0gsca 31082 lbslsat 31102 dimkerim 31111 fedgmullem2 31114 2zrng0 44562 |
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