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Mirrors > Home > MPE Home > Th. List > grpsubeq0 | Structured version Visualization version GIF version |
Description: If the difference between two group elements is zero, they are equal. (subeq0 11432 analog.) (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
grpsubid.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubid.o | ⊢ 0 = (0g‘𝐺) |
grpsubid.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubeq0 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubid.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2733 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2733 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | grpsubid.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 18801 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
6 | 5 | 3adant1 1131 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
7 | 6 | eqeq1d 2735 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) = 0 ↔ (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = 0 )) |
8 | simp1 1137 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Grp) | |
9 | 1, 3 | grpinvcl 18803 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
10 | 9 | 3adant2 1132 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
11 | simp2 1138 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
12 | grpsubid.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
13 | 1, 2, 12, 3 | grpinvid2 18808 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = 0 )) |
14 | 8, 10, 11, 13 | syl3anc 1372 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = 0 )) |
15 | 1, 3 | grpinvinv 18819 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
16 | 15 | 3adant2 1132 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
17 | 16 | eqeq1d 2735 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ 𝑌 = 𝑋)) |
18 | eqcom 2740 | . . 3 ⊢ (𝑌 = 𝑋 ↔ 𝑋 = 𝑌) | |
19 | 17, 18 | bitrdi 287 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ 𝑋 = 𝑌)) |
20 | 7, 14, 19 | 3bitr2d 307 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 Grpcgrp 18753 invgcminusg 18754 -gcsg 18755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 |
This theorem is referenced by: ghmeqker 19040 ghmf1 19042 odcong 19336 subgdisj1 19478 dprdf11 19807 kerf1ghm 20184 lmodsubeq0 20396 lvecvscan2 20589 ip2eq 21073 mdetuni0 21986 tgphaus 23484 nrmmetd 23946 ply1divmo 25516 dvdsq1p 25541 dvdsr1p 25542 ply1remlem 25543 ig1peu 25552 dchr2sum 26637 fermltlchr 32201 znfermltl 32202 linds2eq 32216 eqlkr 37607 hdmap11 40357 hdmapinvlem4 40430 isdomn4 40670 idomrootle 41565 lidldomn1 46305 |
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