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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 10186 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 2610 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2610 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | grpsubid.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 17288 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
6 | 5 | 3adant1 1072 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
7 | 6 | eqeq1d 2612 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) = 0 ↔ (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = 0 )) |
8 | simp1 1054 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Grp) | |
9 | 1, 3 | grpinvcl 17290 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
10 | 9 | 3adant2 1073 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
11 | simp2 1055 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
12 | grpsubid.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
13 | 1, 2, 12, 3 | grpinvid2 17294 | . . 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 1318 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = 0 )) |
15 | 1, 3 | grpinvinv 17305 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
16 | 15 | 3adant2 1073 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
17 | 16 | eqeq1d 2612 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ 𝑌 = 𝑋)) |
18 | eqcom 2617 | . . 3 ⊢ (𝑌 = 𝑋 ↔ 𝑋 = 𝑌) | |
19 | 17, 18 | syl6bb 275 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ 𝑋 = 𝑌)) |
20 | 7, 14, 19 | 3bitr2d 295 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 -gcsg 17247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 |
This theorem is referenced by: ghmeqker 17510 ghmf1 17512 odcong 17791 subgdisj1 17927 dprdf11 18245 kerf1hrm 18566 lmodsubeq0 18745 lvecvscan2 18933 ip2eq 19817 mdetuni0 20246 tgphaus 21730 nrmmetd 22189 ply1divmo 23699 dvdsq1p 23724 dvdsr1p 23725 ply1remlem 23726 ig1peu 23735 dchr2sum 24798 eqlkr 33404 hdmap11 36158 hdmapinvlem4 36231 idomrootle 36792 lidldomn1 41711 |
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