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Mirrors > Home > MPE Home > Th. List > grpsubsub | Structured version Visualization version GIF version |
Description: Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
grpsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubadd.p | ⊢ + = (+g‘𝐺) |
grpsubadd.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubsub | ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + (𝑍 − 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1191 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
2 | grpsubadd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpsubadd.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
4 | 2, 3 | grpsubcl 18948 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵) |
5 | 4 | 3adant3r1 1179 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 − 𝑍) ∈ 𝐵) |
6 | grpsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
7 | eqid 2726 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
8 | 2, 6, 7, 3 | grpsubval 18915 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑌 − 𝑍) ∈ 𝐵) → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + ((invg‘𝐺)‘(𝑌 − 𝑍)))) |
9 | 1, 5, 8 | syl2anc 583 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + ((invg‘𝐺)‘(𝑌 − 𝑍)))) |
10 | 2, 3, 7 | grpinvsub 18950 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘(𝑌 − 𝑍)) = (𝑍 − 𝑌)) |
11 | 10 | 3adant3r1 1179 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘(𝑌 − 𝑍)) = (𝑍 − 𝑌)) |
12 | 11 | oveq2d 7421 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 + ((invg‘𝐺)‘(𝑌 − 𝑍))) = (𝑋 + (𝑍 − 𝑌))) |
13 | 9, 12 | eqtrd 2766 | 1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + (𝑍 − 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 Grpcgrp 18863 invgcminusg 18864 -gcsg 18865 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 |
This theorem is referenced by: ablsubsub 19737 |
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