Theorem linvh 42921

Description: If an element has a unique left inverse, then the value satisfies the left inverse value equation. (Contributed by metakunt, 25-Apr-2025.)

Hypotheses

Ref	Expression
linvh.1	⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))
linvh.2	⊢ (𝜑 → ∃!𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅))

Assertion

Ref	Expression
linvh	⊢ (𝜑 → (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅))

Distinct variable groups:   𝑅,𝑖   𝑖,𝑋

Allowed substitution hint:   𝜑(𝑖)

Proof of Theorem linvh

Step	Hyp	Ref	Expression
1		linvh.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))
2		eqid 2765	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2765	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2765	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2765	. . . . 5 ⊢ (invg‘𝑅) = (invg‘𝑅)
6	2, 3, 4, 5	grpinvval 19091	. . . 4 ⊢ (𝑋 ∈ (Base‘𝑅) → ((invg‘𝑅)‘𝑋) = (℩𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)))
7	1, 6	syl 18	. . 3 ⊢ (𝜑 → ((invg‘𝑅)‘𝑋) = (℩𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)))
8		linvh.2	. . . 4 ⊢ (𝜑 → ∃!𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅))
9		riotacl2 7392	. . . 4 ⊢ (∃!𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅) → (℩𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
10	8, 9	syl 18	. . 3 ⊢ (𝜑 → (℩𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
11	7, 10	eqeltrd 2865	. 2 ⊢ (𝜑 → ((invg‘𝑅)‘𝑋) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
12		oveq1 7426	. . . . 5 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → (𝑖(+g‘𝑅)𝑋) = (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋))
13	12	eqeq1d 2767	. . . 4 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → ((𝑖(+g‘𝑅)𝑋) = (0g‘𝑅) ↔ (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅)))
14	13	elrab 3652	. . 3 ⊢ (((invg‘𝑅)‘𝑋) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)} ↔ (((invg‘𝑅)‘𝑋) ∈ (Base‘𝑅) ∧ (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅)))
15	14	simprbi 503	. 2 ⊢ (((invg‘𝑅)‘𝑋) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)} → (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅))
16	11, 15	syl 18	1 ⊢ (𝜑 → (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∃!wreu 3369  {crab 3418  ‘cfv 6540  ℩crio 7375  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  0_gc0g 17514  inv_gcminusg 19045

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7376  df-ov 7422  df-minusg 19048

This theorem is used by:  primrootsunit1  42922