Theorem mndlrinvb 32973

Description: In a monoid, if an element has both a left-inverse and a right-inverse, they are equal. (Contributed by Thierry Arnoux, 3-Aug-2025.)

Hypotheses

Ref	Expression
mndlrinv.b	⊢ 𝐵 = (Base‘𝐸)
mndlrinv.z	⊢ 0 = (0g‘𝐸)
mndlrinv.p	⊢ + = (+g‘𝐸)
mndlrinv.e	⊢ (𝜑 → 𝐸 ∈ Mnd)
mndlrinv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
mndlrinvb	⊢ (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )))

Distinct variable groups:   𝑢, + ,𝑣   𝑦, +   𝑢, 0 ,𝑣   𝑦, 0   𝑢,𝐵,𝑣   𝑦,𝐵   𝑢,𝑋,𝑣   𝑦,𝑋   𝜑,𝑢,𝑣   𝜑,𝑦

Allowed substitution hints:   𝐸(𝑦,𝑣,𝑢)

Proof of Theorem mndlrinvb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (𝑋 + 𝑧) = (𝑋 + 𝑢))
2	1	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → ((𝑋 + 𝑧) = 0 ↔ (𝑋 + 𝑢) = 0 ))
3		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (𝑧 + 𝑋) = (𝑢 + 𝑋))
4	3	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → ((𝑧 + 𝑋) = 0 ↔ (𝑢 + 𝑋) = 0 ))
5	2, 4	anbi12d 632	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ) ↔ ((𝑋 + 𝑢) = 0 ∧ (𝑢 + 𝑋) = 0 )))
6		simplr 768	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑢 ∈ 𝐵)
7		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑋 + 𝑢) = 0 )
8		mndlrinv.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐸)
9		mndlrinv.z	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐸)
10		mndlrinv.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐸)
11		mndlrinv.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ Mnd)
12	11	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝐸 ∈ Mnd)
13		mndlrinv.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	13	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑋 ∈ 𝐵)
15		simpllr 775	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑣 ∈ 𝐵)
16		simp-4r 783	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑣 + 𝑋) = 0 )
17	8, 9, 10, 12, 14, 15, 6, 16, 7	mndlrinv 32972	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑣 = 𝑢)
18	17	oveq1d 7405	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑣 + 𝑋) = (𝑢 + 𝑋))
19	18, 16	eqtr3d 2767	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑢 + 𝑋) = 0 )
20	7, 19	jca 511	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → ((𝑋 + 𝑢) = 0 ∧ (𝑢 + 𝑋) = 0 ))
21	5, 6, 20	rspcedvdw 3594	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
22	21	r19.29an 3138	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
23	22	an42ds 32386	. . . . 5 ⊢ ((((𝜑 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
24	23	r19.29an 3138	. . . 4 ⊢ (((𝜑 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ) ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
25	24	anasss 466	. . 3 ⊢ ((𝜑 ∧ (∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 )) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
26		oveq2 7398	. . . . . . 7 ⊢ (𝑢 = 𝑧 → (𝑋 + 𝑢) = (𝑋 + 𝑧))
27	26	eqeq1d 2732	. . . . . 6 ⊢ (𝑢 = 𝑧 → ((𝑋 + 𝑢) = 0 ↔ (𝑋 + 𝑧) = 0 ))
28		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → 𝑧 ∈ 𝐵)
29		simprl 770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (𝑋 + 𝑧) = 0 )
30	27, 28, 29	rspcedvdw 3594	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )
31		oveq1 7397	. . . . . . 7 ⊢ (𝑣 = 𝑧 → (𝑣 + 𝑋) = (𝑧 + 𝑋))
32	31	eqeq1d 2732	. . . . . 6 ⊢ (𝑣 = 𝑧 → ((𝑣 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
33		simprr 772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (𝑧 + 𝑋) = 0 )
34	32, 28, 33	rspcedvdw 3594	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 )
35	30, 34	jca 511	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ))
36	35	r19.29an 3138	. . 3 ⊢ ((𝜑 ∧ ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ))
37	25, 36	impbida 800	. 2 ⊢ (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) ↔ ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )))
38		oveq2 7398	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑋 + 𝑦) = (𝑋 + 𝑧))
39	38	eqeq1d 2732	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑋 + 𝑦) = 0 ↔ (𝑋 + 𝑧) = 0 ))
40		oveq1 7397	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
41	40	eqeq1d 2732	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
42	39, 41	anbi12d 632	. . 3 ⊢ (𝑦 = 𝑧 → (((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ) ↔ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )))
43	42	cbvrexvw 3217	. 2 ⊢ (∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ) ↔ ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
44	37, 43	bitr4di 289	1 ⊢ (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Mndcmnd 18668

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-riota 7347  df-ov 7393  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669

This theorem is referenced by:  mndractf1o  32979  isunit3  33199