Theorem linvh 42891

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 2762	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2762	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2762	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2762	. . . . 5 ⊢ (invg‘𝑅) = (invg‘𝑅)
6	2, 3, 4, 5	grpinvval 19053	. . . 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 7385	. . . 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 2862	. 2 ⊢ (𝜑 → ((invg‘𝑅)‘𝑋) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
12		oveq1 7419	. . . . 5 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → (𝑖(+g‘𝑅)𝑋) = (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋))
13	12	eqeq1d 2764	. . . 4 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → ((𝑖(+g‘𝑅)𝑋) = (0g‘𝑅) ↔ (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅)))
14	13	elrab 3649	. . 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 502	. 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 1569   ∈ wcel 2142  ∃!wreu 3366  {crab 3415  ‘cfv 6536  ℩crio 7368  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  0_gc0g 17498  inv_gcminusg 19007

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-riota 7369  df-ov 7415  df-minusg 19010

This theorem is used by:  primrootsunit1  42892