Theorem mndractf1 33120

Description: If an element 𝑋 of a monoid 𝐸 is right-invertible, with inverse 𝑌, then its left-translation 𝐺 is injective. See also grplactf1o 18986. Remark in chapter I. of [BourbakiAlg1] p. 17 . (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)
mndractf1.1	⊢ (𝜑 → 𝑌 ∈ 𝐵)
mndractf1.2	⊢ (𝜑 → (𝑋 + 𝑌) = 0 )

Assertion

Ref	Expression
mndractf1	⊢ (𝜑 → 𝐺:𝐵–1-1→𝐵)

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

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

Proof of Theorem mndractf1

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

Step	Hyp	Ref	Expression
1		mndractfo.b	. . . 4 ⊢ 𝐵 = (Base‘𝐸)
2		mndractfo.p	. . . 4 ⊢ + = (+g‘𝐸)
3		mndractfo.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Mnd)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐸 ∈ Mnd)
5		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
6		mndractfo.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐵)
8	1, 2, 4, 5, 7	mndcld 33114	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 + 𝑋) ∈ 𝐵)
9		mndractfo.f	. . 3 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋))
10	8, 9	fmptd 7068	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
11		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝐺‘𝑖) = (𝐺‘𝑗))
12		oveq1 7375	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑖 → (𝑎 + 𝑋) = (𝑖 + 𝑋))
13		simpllr 776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → 𝑖 ∈ 𝐵)
14		ovexd 7403	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑖 + 𝑋) ∈ V)
15	9, 12, 13, 14	fvmptd3 6973	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝐺‘𝑖) = (𝑖 + 𝑋))
16		oveq1 7375	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑗 → (𝑎 + 𝑋) = (𝑗 + 𝑋))
17		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → 𝑗 ∈ 𝐵)
18		ovexd 7403	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑗 + 𝑋) ∈ V)
19	9, 16, 17, 18	fvmptd3 6973	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝐺‘𝑗) = (𝑗 + 𝑋))
20	11, 15, 19	3eqtr3d 2780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑖 + 𝑋) = (𝑗 + 𝑋))
21	20	oveq1d 7383	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → ((𝑖 + 𝑋) + 𝑌) = ((𝑗 + 𝑋) + 𝑌))
22	3	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → 𝐸 ∈ Mnd)
23	6	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → 𝑋 ∈ 𝐵)
24		mndractf1.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25	24	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → 𝑌 ∈ 𝐵)
26	1, 2, 22, 13, 23, 25	mndassd 33115	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → ((𝑖 + 𝑋) + 𝑌) = (𝑖 + (𝑋 + 𝑌)))
27	1, 2, 22, 17, 23, 25	mndassd 33115	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → ((𝑗 + 𝑋) + 𝑌) = (𝑗 + (𝑋 + 𝑌)))
28	21, 26, 27	3eqtr3d 2780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑖 + (𝑋 + 𝑌)) = (𝑗 + (𝑋 + 𝑌)))
29		mndractf1.2	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 + 𝑌) = 0 )
30	29	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑋 + 𝑌) = 0 )
31	30	oveq2d 7384	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑖 + (𝑋 + 𝑌)) = (𝑖 + 0 ))
32	30	oveq2d 7384	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑗 + (𝑋 + 𝑌)) = (𝑗 + 0 ))
33	28, 31, 32	3eqtr3d 2780	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑖 + 0 ) = (𝑗 + 0 ))
34		mndractfo.z	. . . . . . . 8 ⊢ 0 = (0g‘𝐸)
35	1, 2, 34	mndrid 18692	. . . . . . 7 ⊢ ((𝐸 ∈ Mnd ∧ 𝑖 ∈ 𝐵) → (𝑖 + 0 ) = 𝑖)
36	22, 13, 35	syl2anc 585	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑖 + 0 ) = 𝑖)
37	1, 2, 34	mndrid 18692	. . . . . . 7 ⊢ ((𝐸 ∈ Mnd ∧ 𝑗 ∈ 𝐵) → (𝑗 + 0 ) = 𝑗)
38	22, 17, 37	syl2anc 585	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → (𝑗 + 0 ) = 𝑗)
39	33, 36, 38	3eqtr3d 2780	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) ∧ (𝐺‘𝑖) = (𝐺‘𝑗)) → 𝑖 = 𝑗)
40	39	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐵) ∧ 𝑗 ∈ 𝐵) → ((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗))
41	40	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵)) → ((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗))
42	41	ralrimivva 3181	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝐵 ∀𝑗 ∈ 𝐵 ((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗))
43		dff13 7210	. 2 ⊢ (𝐺:𝐵–1-1→𝐵 ↔ (𝐺:𝐵⟶𝐵 ∧ ∀𝑖 ∈ 𝐵 ∀𝑗 ∈ 𝐵 ((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))
44	10, 42, 43	sylanbrc 584	1 ⊢ (𝜑 → 𝐺:𝐵–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ↦ cmpt 5181  ⟶wf 6496  –1-1→wf1 6497  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  Mndcmnd 18671

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508  df-riota 7325  df-ov 7371  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672

This theorem is referenced by:  mndractf1o  33123