Theorem grpinv11OLD 18982

Description: Obsolete version of grpinv11 18981 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 6834	. . . . 5 ⊢ ((𝑁‘𝑋) = (𝑁‘𝑌) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌)))
2	1	adantl 482	. . . 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 18979	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋)
8	3, 4, 7	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = 𝑋)
10		grpinv11.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11	5, 6	grpinvinv 18979	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌)
12	3, 10, 11	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌)
13	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑌)) = 𝑌)
14	2, 9, 13	3eqtr3d 2783	. . 3 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → 𝑋 = 𝑌)
15	14	ex 413	. 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) → 𝑋 = 𝑌))
16		fveq2 6834	. 2 ⊢ (𝑋 = 𝑌 → (𝑁‘𝑋) = (𝑁‘𝑌))
17	15, 16	impbid1 226	1 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  Basecbs 17177  Grpcgrp 18907  inv_gcminusg 18908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7320  df-ov 7366  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911

This theorem is referenced by: (None)