Theorem mndlactfo 33087

Description: An element 𝑋 of a monoid 𝐸 is left-invertible iff its left-translation 𝐹 is surjective. See also grplactf1o 19020. (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 484	. . . 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 18717	. . . . . 6 ⊢ (𝐸 ∈ Mnd → 0 ∈ 𝐵)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → 0 ∈ 𝐵)
8		foelcdmi 6901	. . . 4 ⊢ ((𝐹:𝐵–onto→𝐵 ∧ 0 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 0 )
9	1, 7, 8	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 0 )
10		mndlactfo.f	. . . . . . 7 ⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))
11		oveq2 7375	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑋 + 𝑎) = (𝑋 + 𝑦))
12		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
13		ovexd 7402	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑋 + 𝑦) ∈ V)
14	10, 11, 12, 13	fvmptd3 6971	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) = (𝑋 + 𝑦))
15	14	eqeq1d 2738	. . . . 5 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦) = 0 ↔ (𝑋 + 𝑦) = 0 ))
16	15	biimpd 229	. . . 4 ⊢ (((𝜑 ∧ 𝐹:𝐵–onto→𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦) = 0 → (𝑋 + 𝑦) = 0 ))
17	16	reximdva 3150	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 0 → ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 ))
18	9, 17	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐹:𝐵–onto→𝐵) → ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 )
19		mndlactfo.p	. . . . . . 7 ⊢ + = (+g‘𝐸)
20	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐸 ∈ Mnd)
21		mndlactfo.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐵)
23		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
24	3, 19, 20, 22, 23	mndcld 33082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋 + 𝑎) ∈ 𝐵)
25	24, 10	fmptd 7066	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐵)
26	25	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) → 𝐹:𝐵⟶𝐵)
27		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝐹‘𝑥) = (𝐹‘(𝑦 + 𝑧)))
28	27	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑧 = (𝐹‘𝑥) ↔ 𝑧 = (𝐹‘(𝑦 + 𝑧))))
29	2	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝐸 ∈ Mnd)
30		simpllr 776	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
31		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
32	3, 19, 29, 30, 31	mndcld 33082	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑦 + 𝑧) ∈ 𝐵)
33	21	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	3, 19, 29, 33, 30, 31	mndassd 33083	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑋 + 𝑦) + 𝑧) = (𝑋 + (𝑦 + 𝑧)))
35		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑋 + 𝑦) = 0 )
36	35	oveq1d 7382	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑋 + 𝑦) + 𝑧) = ( 0 + 𝑧))
37	3, 19, 4	mndlid 18722	. . . . . . . . 9 ⊢ ((𝐸 ∈ Mnd ∧ 𝑧 ∈ 𝐵) → ( 0 + 𝑧) = 𝑧)
38	29, 31, 37	syl2anc 585	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ( 0 + 𝑧) = 𝑧)
39	36, 38	eqtr2d 2772	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = ((𝑋 + 𝑦) + 𝑧))
40		oveq2 7375	. . . . . . . 8 ⊢ (𝑎 = (𝑦 + 𝑧) → (𝑋 + 𝑎) = (𝑋 + (𝑦 + 𝑧)))
41		ovexd 7402	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑋 + (𝑦 + 𝑧)) ∈ V)
42	10, 40, 32, 41	fvmptd3 6971	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝐹‘(𝑦 + 𝑧)) = (𝑋 + (𝑦 + 𝑧)))
43	34, 39, 42	3eqtr4d 2781	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐹‘(𝑦 + 𝑧)))
44	28, 32, 43	rspcedvdw 3567	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) ∧ 𝑧 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 𝑧 = (𝐹‘𝑥))
45	44	ralrimiva 3129	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐹‘𝑥))
46		dffo3 7054	. . . 4 ⊢ (𝐹:𝐵–onto→𝐵 ↔ (𝐹:𝐵⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐹‘𝑥)))
47	26, 45, 46	sylanbrc 584	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑋 + 𝑦) = 0 ) → 𝐹:𝐵–onto→𝐵)
48	47	r19.29an 3141	. 2 ⊢ ((𝜑 ∧ ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 ) → 𝐹:𝐵–onto→𝐵)
49	18, 48	impbida 801	1 ⊢ (𝜑 → (𝐹:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ↦ cmpt 5166  ⟶wf 6494  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Mndcmnd 18702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-riota 7324  df-ov 7370  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703

This theorem is referenced by:  mndlactf1o  33090