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Mirrors > Home > MPE Home > Th. List > grpaddsubass | Structured version Visualization version GIF version |
Description: Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.) |
Ref | Expression |
---|---|
grpsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubadd.p | ⊢ + = (+g‘𝐺) |
grpsubadd.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpaddsubass | ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = (𝑋 + (𝑌 − 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐺 ∈ Grp) | |
2 | simpr1 1195 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
3 | simpr2 1196 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
4 | grpsubadd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | eqid 2733 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
6 | 4, 5 | grpinvcl 18803 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵) |
7 | 6 | 3ad2antr3 1191 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵) |
8 | grpsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | 4, 8 | grpass 18762 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑌) + ((invg‘𝐺)‘𝑍)) = (𝑋 + (𝑌 + ((invg‘𝐺)‘𝑍)))) |
10 | 1, 2, 3, 7, 9 | syl13anc 1373 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + ((invg‘𝐺)‘𝑍)) = (𝑋 + (𝑌 + ((invg‘𝐺)‘𝑍)))) |
11 | 4, 8 | grpcl 18761 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
12 | 11 | 3adant3r3 1185 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 + 𝑌) ∈ 𝐵) |
13 | simpr3 1197 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
14 | grpsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
15 | 4, 8, 5, 14 | grpsubval 18801 | . . 3 ⊢ (((𝑋 + 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑍) = ((𝑋 + 𝑌) + ((invg‘𝐺)‘𝑍))) |
16 | 12, 13, 15 | syl2anc 585 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = ((𝑋 + 𝑌) + ((invg‘𝐺)‘𝑍))) |
17 | 4, 8, 5, 14 | grpsubval 18801 | . . . 4 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) = (𝑌 + ((invg‘𝐺)‘𝑍))) |
18 | 3, 13, 17 | syl2anc 585 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 − 𝑍) = (𝑌 + ((invg‘𝐺)‘𝑍))) |
19 | 18 | oveq2d 7374 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 + (𝑌 − 𝑍)) = (𝑋 + (𝑌 + ((invg‘𝐺)‘𝑍)))) |
20 | 10, 16, 19 | 3eqtr4d 2783 | 1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = (𝑋 + (𝑌 − 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 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: grppncan 18843 grpnpncan 18847 nsgconj 18966 conjghm 19044 conjnmz 19047 conjnmzb 19048 sylow3lem1 19414 sylow3lem2 19415 abladdsub 19598 ablsubsub 19601 cpmadugsumlemF 22241 archiabllem2a 32079 |
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