Theorem mndlrinvb 33157

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 7393	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (𝑋 + 𝑧) = (𝑋 + 𝑢))
2	1	eqeq1d 2758	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → ((𝑋 + 𝑧) = 0 ↔ (𝑋 + 𝑢) = 0 ))
3		oveq1 7392	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (𝑧 + 𝑋) = (𝑢 + 𝑋))
4	3	eqeq1d 2758	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → ((𝑧 + 𝑋) = 0 ↔ (𝑢 + 𝑋) = 0 ))
5	2, 4	anbi12d 640	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ) ↔ ((𝑋 + 𝑢) = 0 ∧ (𝑢 + 𝑋) = 0 )))
6		simplr 776	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑢 ∈ 𝐵)
7		simpr 487	. . . . . . . . 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 740	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝐸 ∈ Mnd)
13		mndlrinv.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	13	ad4antr 740	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑋 ∈ 𝐵)
15		simpllr 783	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑣 ∈ 𝐵)
16		simp-4r 791	. . . . . . . . . . . 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 33156	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → 𝑣 = 𝑢)
18	17	oveq1d 7400	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑣 + 𝑋) = (𝑢 + 𝑋))
19	18, 16	eqtr3d 2793	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → (𝑢 + 𝑋) = 0 )
20	7, 19	jca 518	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → ((𝑋 + 𝑢) = 0 ∧ (𝑢 + 𝑋) = 0 ))
21	5, 6, 20	rspcedvdw 3579	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) ∧ (𝑋 + 𝑢) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
22	21	r19.29an 3160	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 + 𝑋) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
23	22	an42ds 1504	. . . . 5 ⊢ ((((𝜑 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ) ∧ 𝑣 ∈ 𝐵) ∧ (𝑣 + 𝑋) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
24	23	r19.29an 3160	. . . 4 ⊢ (((𝜑 ∧ ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ) ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
25	24	anasss 469	. . 3 ⊢ ((𝜑 ∧ (∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 )) → ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
26		oveq2 7393	. . . . . . 7 ⊢ (𝑢 = 𝑧 → (𝑋 + 𝑢) = (𝑋 + 𝑧))
27	26	eqeq1d 2758	. . . . . 6 ⊢ (𝑢 = 𝑧 → ((𝑋 + 𝑢) = 0 ↔ (𝑋 + 𝑧) = 0 ))
28		simplr 776	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → 𝑧 ∈ 𝐵)
29		simprl 778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (𝑋 + 𝑧) = 0 )
30	27, 28, 29	rspcedvdw 3579	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → ∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 )
31		oveq1 7392	. . . . . . 7 ⊢ (𝑣 = 𝑧 → (𝑣 + 𝑋) = (𝑧 + 𝑋))
32	31	eqeq1d 2758	. . . . . 6 ⊢ (𝑣 = 𝑧 → ((𝑣 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
33		simprr 780	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (𝑧 + 𝑋) = 0 )
34	32, 28, 33	rspcedvdw 3579	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 )
35	30, 34	jca 518	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ))
36	35	r19.29an 3160	. . 3 ⊢ ((𝜑 ∧ ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )) → (∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ))
37	25, 36	impbida 808	. 2 ⊢ (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) ↔ ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )))
38		oveq2 7393	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑋 + 𝑦) = (𝑋 + 𝑧))
39	38	eqeq1d 2758	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑋 + 𝑦) = 0 ↔ (𝑋 + 𝑧) = 0 ))
40		oveq1 7392	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
41	40	eqeq1d 2758	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
42	39, 41	anbi12d 640	. . 3 ⊢ (𝑦 = 𝑧 → (((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ) ↔ ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 )))
43	42	cbvrexvw 3235	. 2 ⊢ (∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ) ↔ ∃𝑧 ∈ 𝐵 ((𝑋 + 𝑧) = 0 ∧ (𝑧 + 𝑋) = 0 ))
44	37, 43	bitr4di 291	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 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∃wrex 3080  ‘cfv 6510  (class class class)co 7385  Basecbs 17221  +_gcplusg 17262  0_gc0g 17444  Mndcmnd 18744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6512  df-fv 6518  df-riota 7342  df-ov 7388  df-0g 17446  df-mgm 18650  df-sgrp 18729  df-mnd 18745

This theorem is referenced by:  mndractf1o  33163  isunit3  33375