Theorem grpinv11OLD 18996

Description: Obsolete version of grpinv11 18995 as of 8-Jul-2025. (Contributed by NM, 22-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinvinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinvinv.n	⊢ 𝑁 = (invg‘𝐺)
grpinv11.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grpinv11.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
grpinv11.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
grpinv11OLD	⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌))

Proof of Theorem grpinv11OLD

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . 5 ⊢ ((𝑁‘𝑋) = (𝑁‘𝑌) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌)))
2	1	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌)))
3		grpinv11.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		grpinv11.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		grpinvinv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
6		grpinvinv.n	. . . . . . 7 ⊢ 𝑁 = (invg‘𝐺)
7	5, 6	grpinvinv 18993	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋)
8	3, 4, 7	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = 𝑋)
10		grpinv11.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11	5, 6	grpinvinv 18993	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌)
12	3, 10, 11	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑌)) = 𝑌)
14	2, 9, 13	3eqtr3d 2779	. . 3 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → 𝑋 = 𝑌)
15	14	ex 412	. 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) → 𝑋 = 𝑌))
16		fveq2 6881	. 2 ⊢ (𝑋 = 𝑌 → (𝑁‘𝑋) = (𝑁‘𝑌))
17	15, 16	impbid1 225	1 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6536  Basecbs 17233  Grpcgrp 18921  inv_gcminusg 18922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-riota 7367  df-ov 7413  df-0g 17460  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-minusg 18925

This theorem is referenced by: (None)