Theorem mndractfo 33019

Description: An element 𝑋 of a monoid 𝐸 is right-invertible iff its right-translation 𝐺 is surjective. (Contributed by Thierry Arnoux, 3-Aug-2025.)

Hypotheses

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

Assertion

Ref	Expression
mndractfo	⊢ (𝜑 → (𝐺:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Distinct variable groups:   + ,𝑎   0 ,𝑎,𝑦   𝐵,𝑎,𝑦   𝐺,𝑎,𝑦   𝑋,𝑎   𝜑,𝑎,𝑦

Allowed substitution hints:   + (𝑦)   𝐸(𝑦,𝑎)   𝑋(𝑦)

Proof of Theorem mndractfo

Dummy variables 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → 𝐺:𝐵–onto→𝐵)
2		mndractfo.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Mnd)
3		mndractfo.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
4		mndractfo.z	. . . . . . 7 ⊢ 0 = (0g‘𝐸)
5	3, 4	mndidcl 18758	. . . . . 6 ⊢ (𝐸 ∈ Mnd → 0 ∈ 𝐵)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → 0 ∈ 𝐵)
8		foelcdmi 6968	. . . 4 ⊢ ((𝐺:𝐵–onto→𝐵 ∧ 0 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 )
9	1, 7, 8	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 )
10		mndractfo.f	. . . . . . 7 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋))
11		oveq1 7436	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 + 𝑋) = (𝑦 + 𝑋))
12		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
13		ovexd 7464	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 + 𝑋) ∈ V)
14	10, 11, 12, 13	fvmptd3 7037	. . . . . 6 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝑦 + 𝑋))
15	14	eqeq1d 2738	. . . . 5 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑦) = 0 ↔ (𝑦 + 𝑋) = 0 ))
16	15	biimpd 229	. . . 4 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑦) = 0 → (𝑦 + 𝑋) = 0 ))
17	16	reximdva 3167	. . 3 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → (∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
18	9, 17	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
19		mndractfo.p	. . . . . . 7 ⊢ + = (+g‘𝐸)
20	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐸 ∈ Mnd)
21		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
22		mndractfo.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐵)
24	3, 19, 20, 21, 23	mndcld 33012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 + 𝑋) ∈ 𝐵)
25	24, 10	fmptd 7132	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
26	25	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵⟶𝐵)
27		fveq2 6904	. . . . . . 7 ⊢ (𝑥 = (𝑧 + 𝑦) → (𝐺‘𝑥) = (𝐺‘(𝑧 + 𝑦)))
28	27	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = (𝑧 + 𝑦) → (𝑧 = (𝐺‘𝑥) ↔ 𝑧 = (𝐺‘(𝑧 + 𝑦))))
29	2	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝐸 ∈ Mnd)
30		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
31		simpllr 776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
32	3, 19, 29, 30, 31	mndcld 33012	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑧 + 𝑦) ∈ 𝐵)
33	22	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	3, 19, 29, 30, 31, 33	mndassd 33013	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑧 + 𝑦) + 𝑋) = (𝑧 + (𝑦 + 𝑋)))
35		oveq1 7436	. . . . . . . 8 ⊢ (𝑎 = (𝑧 + 𝑦) → (𝑎 + 𝑋) = ((𝑧 + 𝑦) + 𝑋))
36		ovexd 7464	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑧 + 𝑦) + 𝑋) ∈ V)
37	10, 35, 32, 36	fvmptd3 7037	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝐺‘(𝑧 + 𝑦)) = ((𝑧 + 𝑦) + 𝑋))
38		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑦 + 𝑋) = 0 )
39	38	oveq2d 7445	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑧 + (𝑦 + 𝑋)) = (𝑧 + 0 ))
40	3, 19, 4	mndrid 18764	. . . . . . . . 9 ⊢ ((𝐸 ∈ Mnd ∧ 𝑧 ∈ 𝐵) → (𝑧 + 0 ) = 𝑧)
41	29, 30, 40	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑧 + 0 ) = 𝑧)
42	39, 41	eqtr2d 2777	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝑧 + (𝑦 + 𝑋)))
43	34, 37, 42	3eqtr4rd 2787	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐺‘(𝑧 + 𝑦)))
44	28, 32, 43	rspcedvdw 3624	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥))
45	44	ralrimiva 3145	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥))
46		dffo3 7120	. . . 4 ⊢ (𝐺:𝐵–onto→𝐵 ↔ (𝐺:𝐵⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥)))
47	26, 45, 46	sylanbrc 583	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵–onto→𝐵)
48	47	r19.29an 3157	. 2 ⊢ ((𝜑 ∧ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵–onto→𝐵)
49	18, 48	impbida 801	1 ⊢ (𝜑 → (𝐺:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ↦ cmpt 5223  ⟶wf 6555  –onto→wfo 6557  ‘cfv 6559  (class class class)co 7429  Basecbs 17243  +_gcplusg 17293  0_gc0g 17480  Mndcmnd 18743

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fo 6565  df-fv 6567  df-riota 7386  df-ov 7432  df-0g 17482  df-mgm 18649  df-sgrp 18728  df-mnd 18744

This theorem is referenced by:  mndractf1o  33021