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Mirrors > Home > MPE Home > Th. List > grppncan | Structured version Visualization version GIF version |
Description: Cancellation law for subtraction (pncan 11463 analog). (Contributed by NM, 16-Apr-2014.) |
Ref | Expression |
---|---|
grpsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubadd.p | ⊢ + = (+g‘𝐺) |
grpsubadd.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grppncan | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ Grp) | |
2 | simp2 1134 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | simp3 1135 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
4 | grpsubadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | grpsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
6 | grpsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
7 | 4, 5, 6 | grpaddsubass 18948 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑌) = (𝑋 + (𝑌 − 𝑌))) |
8 | 1, 2, 3, 3, 7 | syl13anc 1369 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑌) = (𝑋 + (𝑌 − 𝑌))) |
9 | eqid 2724 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
10 | 4, 9, 6 | grpsubid 18942 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 − 𝑌) = (0g‘𝐺)) |
11 | 10 | oveq2d 7417 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝑌 − 𝑌)) = (𝑋 + (0g‘𝐺))) |
12 | 11 | 3adant2 1128 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝑌 − 𝑌)) = (𝑋 + (0g‘𝐺))) |
13 | 4, 5, 9 | grprid 18888 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋) |
14 | 13 | 3adant3 1129 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋) |
15 | 8, 12, 14 | 3eqtrd 2768 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 +gcplusg 17196 0gc0g 17384 Grpcgrp 18853 -gcsg 18855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 |
This theorem is referenced by: grpnpcan 18950 grppnpcan2 18952 ssnmz 19083 conjnmz 19167 cntrsubgnsg 19249 sylow2blem3 19532 sylow3lem2 19538 subgdisj1 19601 pgpfac1lem3 19989 lmodvpncan 20751 opnsubg 23934 lfl0 38425 nelsubgcld 41564 |
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