Theorem mndractfo 33638

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 490	. . . 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 18946	. . . . . 6 ⊢ (𝐸 ∈ Mnd → 0 ∈ 𝐵)
6	2, 5	syl 18	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → 0 ∈ 𝐵)
8		foelcdmi 6949	. . . 4 ⊢ ((𝐺:𝐵–onto→𝐵 ∧ 0 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 )
9	1, 7, 8	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 )
10		mndractfo.f	. . . . . . 7 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋))
11		oveq1 7429	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 + 𝑋) = (𝑦 + 𝑋))
12		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
13		ovexd 7457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 + 𝑋) ∈ V)
14	10, 11, 12, 13	fvmptd3 7020	. . . . . 6 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝑦 + 𝑋))
15	14	eqeq1d 2762	. . . . 5 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑦) = 0 ↔ (𝑦 + 𝑋) = 0 ))
16	15	biimpd 232	. . . 4 ⊢ (((𝜑 ∧ 𝐺:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑦) = 0 → (𝑦 + 𝑋) = 0 ))
17	16	reximdva 3175	. . 3 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → (∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
18	9, 17	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝐺:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
19		mndractfo.p	. . . . . . 7 ⊢ + = (+g‘𝐸)
20	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐸 ∈ Mnd)
21		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
22		mndractfo.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23	22	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐵)
24	3, 19, 20, 21, 23	mndcld 33631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 + 𝑋) ∈ 𝐵)
25	24, 10	fmptd 7117	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
26	25	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵⟶𝐵)
27		fveq2 6888	. . . . . . 7 ⊢ (𝑥 = (𝑧 + 𝑦) → (𝐺‘𝑥) = (𝐺‘(𝑧 + 𝑦)))
28	27	eqeq2d 2771	. . . . . 6 ⊢ (𝑥 = (𝑧 + 𝑦) → (𝑧 = (𝐺‘𝑥) ↔ 𝑧 = (𝐺‘(𝑧 + 𝑦))))
29	2	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝐸 ∈ Mnd)
30		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
31		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
32	3, 19, 29, 30, 31	mndcld 33631	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑧 + 𝑦) ∈ 𝐵)
33	22	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	3, 19, 29, 30, 31, 33	mndassd 33632	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑧 + 𝑦) + 𝑋) = (𝑧 + (𝑦 + 𝑋)))
35		oveq1 7429	. . . . . . . 8 ⊢ (𝑎 = (𝑧 + 𝑦) → (𝑎 + 𝑋) = ((𝑧 + 𝑦) + 𝑋))
36		ovexd 7457	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑧 + 𝑦) + 𝑋) ∈ V)
37	10, 35, 32, 36	fvmptd3 7020	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝐺‘(𝑧 + 𝑦)) = ((𝑧 + 𝑦) + 𝑋))
38		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑦 + 𝑋) = 0 )
39	38	oveq2d 7438	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑧 + (𝑦 + 𝑋)) = (𝑧 + 0 ))
40	3, 19, 4	mndrid 18952	. . . . . . . . 9 ⊢ ((𝐸 ∈ Mnd ∧ 𝑧 ∈ 𝐵) → (𝑧 + 0 ) = 𝑧)
41	29, 30, 40	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑧 + 0 ) = 𝑧)
42	39, 41	eqtr2d 2796	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝑧 + (𝑦 + 𝑋)))
43	34, 37, 42	3eqtr4rd 2806	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐺‘(𝑧 + 𝑦)))
44	28, 32, 43	rspcedvdw 3579	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥))
45	44	ralrimiva 3154	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥))
46		dffo3 7105	. . . 4 ⊢ (𝐺:𝐵–onto→𝐵 ↔ (𝐺:𝐵⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥)))
47	26, 45, 46	sylanbrc 595	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵–onto→𝐵)
48	47	r19.29an 3166	. 2 ⊢ ((𝜑 ∧ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵–onto→𝐵)
49	18, 48	impbida 813	1 ⊢ (𝜑 → (𝐺:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ↦ cmpt 5185  ⟶wf 6537  –onto→wfo 6539  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  +_gcplusg 17435  0_gc0g 17617  Mndcmnd 18930

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-riota 7379  df-ov 7425  df-0g 17619  df-mgm 18823  df-sgrp 18915  df-mnd 18931

This theorem is used by:  mndractf1o  33640