Theorem mndlactfo 33634

Description: An element 𝑋 of a monoid 𝐸 is left-invertible iff its left-translation 𝐹 is surjective. See also grplactf1o 19259. (Contributed by Thierry Arnoux, 3-Aug-2025.)

Hypotheses

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

Assertion

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

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

Allowed substitution hints:   𝐸(𝑦, 𝑎)

Proof of Theorem mndlactfo

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

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → 𝐹:𝐵–onto→𝐵)
2		mndlactfo.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Mnd)
3		mndlactfo.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
4		mndlactfo.z	. . . . . . 7 ⊢ 0 = (0g‘𝐸)
5	3, 4	mndidcl 18944	. . . . . 6 ⊢ (𝐸 ∈ Mnd → 0 ∈ 𝐵)
6	2, 5	syl 18	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → 0 ∈ 𝐵)
8		foelcdmi 6946	. . . 4 ⊢ ((𝐹:𝐵–onto→𝐵 ∧ 0 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 0 )
9	1, 7, 8	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 0 )
10		mndlactfo.f	. . . . . . 7 ⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))
11		oveq2 7428	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑋 + 𝑎) = (𝑋 + 𝑦))
12		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
13		ovexd 7455	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑋 + 𝑦) ∈ V)
14	10, 11, 12, 13	fvmptd3 7017	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) = (𝑋 + 𝑦))
15	14	eqeq1d 2762	. . . . 5 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦) = 0 ↔ (𝑋 + 𝑦) = 0 ))
16	15	biimpd 232	. . . 4 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦) = 0 → (𝑋 + 𝑦) = 0 ))
17	16	reximdva 3175	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 0 → ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 ))
18	9, 17	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 )
19		mndlactfo.p	. . . . . . 7 ⊢ + = (+g‘𝐸)
20	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐸 ∈ Mnd)
21		mndlactfo.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22	21	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐵)
23		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
24	3, 19, 20, 22, 23	mndcld 33629	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋 + 𝑎) ∈ 𝐵)
25	24, 10	fmptd 7114	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐵)
26	25	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) → 𝐹:𝐵⟶𝐵)
27		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐹‘𝑥) = (𝐹‘(𝑦 + 𝑧)))
28	27	eqeq2d 2771	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑧 = (𝐹‘𝑥) ↔ 𝑧 = (𝐹‘(𝑦 + 𝑧))))
29	2	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝐸 ∈ Mnd)
30		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
31		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
32	3, 19, 29, 30, 31	mndcld 33629	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑦 + 𝑧) ∈ 𝐵)
33	21	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	3, 19, 29, 33, 30, 31	mndassd 33630	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑋 + 𝑦) + 𝑧) = (𝑋 + (𝑦 + 𝑧)))
35		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑋 + 𝑦) = 0 )
36	35	oveq1d 7435	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑋 + 𝑦) + 𝑧) = ( 0 + 𝑧))
37	3, 19, 4	mndlid 18949	. . . . . . . . 9 ⊢ ((𝐸 ∈ Mnd ∧ 𝑧 ∈ 𝐵) → ( 0 + 𝑧) = 𝑧)
38	29, 31, 37	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ( 0 + 𝑧) = 𝑧)
39	36, 38	eqtr2d 2796	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = ((𝑋 + 𝑦) + 𝑧))
40		oveq2 7428	. . . . . . . 8 ⊢ (𝑎 = (𝑦 + 𝑧) → (𝑋 + 𝑎) = (𝑋 + (𝑦 + 𝑧)))
41		ovexd 7455	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑋 + (𝑦 + 𝑧)) ∈ V)
42	10, 40, 32, 41	fvmptd3 7017	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝐹‘(𝑦 + 𝑧)) = (𝑋 + (𝑦 + 𝑧)))
43	34, 39, 42	3eqtr4d 2805	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐹‘(𝑦 + 𝑧)))
44	28, 32, 43	rspcedvdw 3579	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 𝑧 = (𝐹‘𝑥))
45	44	ralrimiva 3154	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐹‘𝑥))
46		dffo3 7102	. . . 4 ⊢ (𝐹:𝐵–onto→𝐵 ↔ (𝐹:𝐵⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐹‘𝑥)))
47	26, 45, 46	sylanbrc 595	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) → 𝐹:𝐵–onto→𝐵)
48	47	r19.29an 3166	. 2 ⊢ ((𝜑 ∧ ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 ) → 𝐹:𝐵–onto→𝐵)
49	18, 48	impbida 813	1 ⊢ (𝜑 → (𝐹:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 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 6534  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  +_gcplusg 17433  0_gc0g 17615  Mndcmnd 18928

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 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-riota 7377  df-ov 7423  df-0g 17617  df-mgm 18821  df-sgrp 18913  df-mnd 18929

This theorem is used by:  mndlactf1o  33637