Theorem mndlactf1 33032

Description: If an element 𝑋 of a monoid 𝐸 is right-invertible, with inverse 𝑌, then its left-translation 𝐹 is injective. See also grplactf1o 19063. Remark in chapter I. of [BourbakiAlg1] p. 17 . (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)
mndlactf1.1	⊢ (𝜑 → 𝑌 ∈ 𝐵)
mndlactf1.2	⊢ (𝜑 → (𝑌 + 𝑋) = 0 )

Assertion

Ref	Expression
mndlactf1	⊢ (𝜑 → 𝐹:𝐵–1-1→𝐵)

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

Allowed substitution hints:   𝐸(𝑎)   𝑌(𝑎)

Proof of Theorem mndlactf1

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndlactfo.b	. . . 4 ⊢ 𝐵 = (Base‘𝐸)
2		mndlactfo.p	. . . 4 ⊢ + = (+g‘𝐸)
3		mndlactfo.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Mnd)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐸 ∈ Mnd)
5		mndlactfo.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐵)
7		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
8	1, 2, 4, 6, 7	mndcld 33028	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋 + 𝑎) ∈ 𝐵)
9		mndlactfo.f	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))
10	8, 9	fmptd 7133	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶𝐵)
11		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝐹‘𝑖) = (𝐹‘𝑗))
12		oveq2 7440	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑖 → (𝑋 + 𝑎) = (𝑋 + 𝑖))
13		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → 𝑖 ∈ 𝐵)
14		ovexd 7467	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝑋 + 𝑖) ∈ V)
15	9, 12, 13, 14	fvmptd3 7038	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝐹‘𝑖) = (𝑋 + 𝑖))
16		oveq2 7440	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑗 → (𝑋 + 𝑎) = (𝑋 + 𝑗))
17		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → 𝑗 ∈ 𝐵)
18		ovexd 7467	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝑋 + 𝑗) ∈ V)
19	9, 16, 17, 18	fvmptd3 7038	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝐹‘𝑗) = (𝑋 + 𝑗))
20	11, 15, 19	3eqtr3d 2784	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝑋 + 𝑖) = (𝑋 + 𝑗))
21	20	oveq2d 7448	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝑌 + (𝑋 + 𝑖)) = (𝑌 + (𝑋 + 𝑗)))
22	3	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → 𝐸 ∈ Mnd)
23		mndlactf1.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
24	23	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → 𝑌 ∈ 𝐵)
25	5	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → 𝑋 ∈ 𝐵)
26	1, 2, 22, 24, 25, 13	mndassd 33029	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ((𝑌 + 𝑋) + 𝑖) = (𝑌 + (𝑋 + 𝑖)))
27	1, 2, 22, 24, 25, 17	mndassd 33029	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ((𝑌 + 𝑋) + 𝑗) = (𝑌 + (𝑋 + 𝑗)))
28	21, 26, 27	3eqtr4d 2786	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ((𝑌 + 𝑋) + 𝑖) = ((𝑌 + 𝑋) + 𝑗))
29		mndlactf1.2	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 + 𝑋) = 0 )
30	29	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → (𝑌 + 𝑋) = 0 )
31	30	oveq1d 7447	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ((𝑌 + 𝑋) + 𝑖) = ( 0 + 𝑖))
32	30	oveq1d 7447	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ((𝑌 + 𝑋) + 𝑗) = ( 0 + 𝑗))
33	28, 31, 32	3eqtr3d 2784	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ( 0 + 𝑖) = ( 0 + 𝑗))
34		mndlactfo.z	. . . . . . . 8 ⊢ 0 = (0g‘𝐸)
35	1, 2, 34	mndlid 18768	. . . . . . 7 ⊢ ((𝐸 ∈ Mnd ∧ 𝑖 ∈ 𝐵) → ( 0 + 𝑖) = 𝑖)
36	22, 13, 35	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ( 0 + 𝑖) = 𝑖)
37	1, 2, 34	mndlid 18768	. . . . . . 7 ⊢ ((𝐸 ∈ Mnd ∧ 𝑗 ∈ 𝐵) → ( 0 + 𝑗) = 𝑗)
38	22, 17, 37	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → ( 0 + 𝑗) = 𝑗)
39	33, 36, 38	3eqtr3d 2784	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐹‘𝑖) = (𝐹‘𝑗)) → 𝑖 = 𝑗)
40	39	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) → ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
41	40	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵)) → ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
42	41	ralrimivva 3201	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝐵 ∀𝑗 ∈ 𝐵 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗))
43		dff13 7276	. 2 ⊢ (𝐹:𝐵–1-1→𝐵 ↔ (𝐹:𝐵⟶𝐵 ∧ ∀𝑖 ∈ 𝐵 ∀𝑗 ∈ 𝐵 ((𝐹‘𝑖) = (𝐹‘𝑗) → 𝑖 = 𝑗)))
44	10, 42, 43	sylanbrc 583	1 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ↦ cmpt 5224  ⟶wf 6556  –1-1→wf1 6557  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  0_gc0g 17485  Mndcmnd 18748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  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 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fv 6568  df-riota 7389  df-ov 7435  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749

This theorem is referenced by:  mndlactf1o  33036