Theorem mndractfo 33689

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⟶onto𝐵 ↔ ∃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⟶onto𝐵) → 𝐺:𝐵⟶onto𝐵)
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 18989	. . . . . 6 ⊢ (𝐸 ∈ Mnd → 0 ∈ 𝐵)
6	2, 5	syl 18	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) → 0 ∈ 𝐵)
8		foelcdmi 6977	. . . 4 ⊢ ((𝐺:𝐵⟶onto𝐵 ∧ 0 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 )
9	1, 7, 8	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) → ∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 )
10		mndractfo.f	. . . . . . 7 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋))
11		oveq1 7458	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 + 𝑋) = (𝑦 + 𝑋))
12		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
13		ovexd 7486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 + 𝑋) ∈ V)
14	10, 11, 12, 13	fvmptd3 7048	. . . . . 6 ⊢ (((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝑦 + 𝑋))
15	14	eqeq1d 2765	. . . . 5 ⊢ (((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑦) = 0 ↔ (𝑦 + 𝑋) = 0 ))
16	15	biimpd 232	. . . 4 ⊢ (((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐺‘𝑦) = 0 → (𝑦 + 𝑋) = 0 ))
17	16	reximdva 3179	. . 3 ⊢ ((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) → (∃𝑦 ∈ 𝐵 (𝐺‘𝑦) = 0 → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
18	9, 17	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝐺:𝐵⟶onto𝐵) → ∃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 33682	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 + 𝑋) ∈ 𝐵)
25	24, 10	fmptd 7146	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
26	25	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵⟶𝐵)
27		fveq2 6916	. . . . . . 7 ⊢ (𝑥 = (𝑧 + 𝑦) → (𝐺‘𝑥) = (𝐺‘(𝑧 + 𝑦)))
28	27	eqeq2d 2774	. . . . . 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 33682	. . . . . 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 33683	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑧 + 𝑦) + 𝑋) = (𝑧 + (𝑦 + 𝑋)))
35		oveq1 7458	. . . . . . . 8 ⊢ (𝑎 = (𝑧 + 𝑦) → (𝑎 + 𝑋) = ((𝑧 + 𝑦) + 𝑋))
36		ovexd 7486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ((𝑧 + 𝑦) + 𝑋) ∈ V)
37	10, 35, 32, 36	fvmptd3 7048	. . . . . . 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 7467	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → (𝑧 + (𝑦 + 𝑋)) = (𝑧 + 0 ))
40	3, 19, 4	mndrid 18995	. . . . . . . . 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 2799	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝑧 + (𝑦 + 𝑋)))
43	34, 37, 42	3eqtr4rd 2809	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐺‘(𝑧 + 𝑦)))
44	28, 32, 43	rspcedvdw 3583	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) ∧ 𝑧 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥))
45	44	ralrimiva 3158	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 + 𝑋) = 0 ) → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑧 = (𝐺‘𝑥))
46		dffo3 7134	. . . 4 ⊢ (𝐺:𝐵⟶onto𝐵 ↔ (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⟶onto𝐵)
48	47	r19.29an 3170	. 2 ⊢ ((𝜑 ∧ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) → 𝐺:𝐵⟶onto𝐵)
49	18, 48	impbida 813	1 ⊢ (𝜑 → (𝐺:𝐵⟶onto𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ↦ cmpt 5192  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  0_gc0g 17660  Mndcmnd 18973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-riota 7408  df-ov 7454  df-0g 17662  df-mgm 18866  df-sgrp 18958  df-mnd 18974

This theorem is used by:  mndractf1o  33691