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Mirrors > Home > MPE Home > Th. List > ablpncan2 | Structured version Visualization version GIF version |
Description: Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.) |
Ref | Expression |
---|---|
ablsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
ablsubadd.p | ⊢ + = (+g‘𝐺) |
ablsubadd.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
ablpncan2 | ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑋) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1131 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Abel) | |
2 | simp2 1132 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | simp3 1133 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
4 | ablsubadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | ablsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
6 | ablsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
7 | 4, 5, 6 | abladdsub 18930 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑋) = ((𝑋 − 𝑋) + 𝑌)) |
8 | 1, 2, 3, 2, 7 | syl13anc 1367 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑋) = ((𝑋 − 𝑋) + 𝑌)) |
9 | ablgrp 18906 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
10 | 1, 9 | syl 17 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Grp) |
11 | eqid 2820 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | 4, 11, 6 | grpsubid 18178 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
13 | 10, 2, 12 | syl2anc 586 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
14 | 13 | oveq1d 7164 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑋) + 𝑌) = ((0g‘𝐺) + 𝑌)) |
15 | 4, 5, 11 | grplid 18128 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌) |
16 | 10, 3, 15 | syl2anc 586 | . 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌) |
17 | 8, 14, 16 | 3eqtrd 2859 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑋) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 0gc0g 16708 Grpcgrp 18098 -gcsg 18100 Abelcabl 18902 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-minusg 18102 df-sbg 18103 df-cmn 18903 df-abl 18904 |
This theorem is referenced by: lssvancl1 19711 lspprabs 19862 lsmcv 19908 ngpocelbl 23308 ttgcontlem1 26669 |
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