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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 12777. (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 12520 | . 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 12512 | . . . . . 6 ⊢ Base Fn V | |
10 | 5 | elexd 2750 | . . . . . 6 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑅 ∈ V) |
11 | funfvex 5531 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V) | |
12 | 11 | funfni 5315 | . . . . . 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 4142 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
16 | 2, 8, 15, 5 | ressplusgd 12579 | . 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 12768 | . . 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 12769 | . . 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 12734 | 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 5210 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 ↾s cress 12455 +gcplusg 12528 0gc0g 12693 Mndcmnd 12749 |
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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-ltadd 7924 |
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 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 |
This theorem is referenced by: submnd0 12777 zring0 13359 |
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