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Mirrors > Home > MPE Home > Th. List > grpsubinv | Structured version Visualization version GIF version |
Description: Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.) |
Ref | Expression |
---|---|
grpsubinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubinv.p | ⊢ + = (+g‘𝐺) |
grpsubinv.m | ⊢ − = (-g‘𝐺) |
grpsubinv.n | ⊢ 𝑁 = (invg‘𝐺) |
grpsubinv.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
grpsubinv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
grpsubinv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
grpsubinv | ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubinv.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | grpsubinv.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
3 | grpsubinv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | grpsubinv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | grpsubinv.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 4, 5 | grpinvcl 17822 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
7 | 2, 3, 6 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵) |
8 | grpsubinv.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | grpsubinv.m | . . . 4 ⊢ − = (-g‘𝐺) | |
10 | 4, 8, 5, 9 | grpsubval 17820 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑁‘𝑌) ∈ 𝐵) → (𝑋 − (𝑁‘𝑌)) = (𝑋 + (𝑁‘(𝑁‘𝑌)))) |
11 | 1, 7, 10 | syl2anc 581 | . 2 ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + (𝑁‘(𝑁‘𝑌)))) |
12 | 4, 5 | grpinvinv 17837 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
13 | 2, 3, 12 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
14 | 13 | oveq2d 6922 | . 2 ⊢ (𝜑 → (𝑋 + (𝑁‘(𝑁‘𝑌))) = (𝑋 + 𝑌)) |
15 | 11, 14 | eqtrd 2862 | 1 ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 +gcplusg 16306 Grpcgrp 17777 invgcminusg 17778 -gcsg 17779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-0g 16456 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-minusg 17781 df-sbg 17782 |
This theorem is referenced by: issubg4 17965 isnsg3 17980 lsmelvalm 18418 ablsub2inv 18570 ablsubsub4 18578 istgp2 22266 nmtri 22801 baerlem5amN 37792 baerlem5abmN 37794 |
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