grpsubinv - Intuitionistic Logic Explorer

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Theorem grpsubinv 13717

Description: Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)

**Hypotheses**
Ref	Expression
grpsubinv.b	⊢ 𝐵 = (Base‘𝐺)
grpsubinv.p	⊢ + = (+_g‘𝐺)
grpsubinv.m	⊢ − = (-_g‘𝐺)
grpsubinv.n	⊢ 𝑁 = (inv_g‘𝐺)
grpsubinv.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grpsubinv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
grpsubinv.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
grpsubinv	⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌))

**Proof of Theorem grpsubinv**
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 = (inv_g‘𝐺)
6	4, 5	grpinvcl 13692	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵)
7	2, 3, 6	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵)
8		grpsubinv.p	. . . 4 ⊢ + = (+_g‘𝐺)
9		grpsubinv.m	. . . 4 ⊢ − = (-_g‘𝐺)
10	4, 8, 5, 9	grpsubval 13690	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑁‘𝑌) ∈ 𝐵) → (𝑋 − (𝑁‘𝑌)) = (𝑋 + (𝑁‘(𝑁‘𝑌))))
11	1, 7, 10	syl2anc 411	. 2 ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + (𝑁‘(𝑁‘𝑌))))
12	4, 5	grpinvinv 13711	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌)
13	2, 3, 12	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌)
14	13	oveq2d 6044	. 2 ⊢ (𝜑 → (𝑋 + (𝑁‘(𝑁‘𝑌))) = (𝑋 + 𝑌))
15	11, 14	eqtrd 2264	1 ⊢ (𝜑 → (𝑋 − (𝑁‘𝑌)) = (𝑋 + 𝑌))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ∈ wcel 2202 ‘cfv 5333 (class class class)co 6028 Basecbs 13143 +_gcplusg 13221 Grpcgrp 13644 inv_gcminusg 13645 -_gcsg 13646

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8166 ax-resscn 8167 ax-1re 8169 ax-addrcl 8172

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-inn 9187 df-2 9245 df-ndx 13146 df-slot 13147 df-base 13149 df-plusg 13234 df-0g 13402 df-mgm 13500 df-sgrp 13546 df-mnd 13561 df-grp 13647 df-minusg 13648 df-sbg 13649

This theorem is referenced by: issubg4m 13841 isnsg3 13855 ablsub2inv 13959 ablsubsub4 13967

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