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| Description: If the difference between two group elements is zero, they are equal. (subeq0 11535 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 2737 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
| 4 | grpsubid.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
| 5 | 1, 2, 3, 4 | grpsubval 19003 | . . . 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 2739 | . 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 19005 | . . . 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 19010 | . . 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 1373 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = 0 )) | 
| 15 | 1, 3 | grpinvinv 19023 | . . . . 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 2739 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑋 ↔ 𝑌 = 𝑋)) | 
| 18 | eqcom 2744 | . . 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 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 invgcminusg 18952 -gcsg 18953 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 | 
| This theorem is referenced by: ghmeqker 19261 ghmf1 19264 kerf1ghm 19265 odcong 19567 subgdisj1 19709 dprdf11 20043 isdomn4 20716 lmodsubeq0 20919 lvecvscan2 21114 fermltlchr 21544 ip2eq 21671 mdetuni0 22627 tgphaus 24125 nrmmetd 24587 ply1divmo 26175 dvdsq1p 26202 dvdsr1p 26203 ply1remlem 26204 idomrootle 26212 ig1peu 26214 dchr2sum 27317 rlocf1 33277 fracerl 33308 znfermltl 33394 linds2eq 33409 assalactf1o 33686 eqlkr 39100 hdmap11 41850 hdmapinvlem4 41923 aks6d1c6lem2 42172 aks6d1c6lem3 42173 lidldomn1 48147 | 
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