Theorem linvh 43412

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 2763	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2763	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2763	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2763	. . . . 5 ⊢ (invg‘𝑅) = (invg‘𝑅)
6	2, 3, 4, 5	grpinvval 19241	. . . 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 7424	. . . 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 2864	. 2 ⊢ (𝜑 → ((invg‘𝑅)‘𝑋) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
12		oveq1 7458	. . . . 5 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → (𝑖(+g‘𝑅)𝑋) = (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋))
13	12	eqeq1d 2765	. . . 4 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → ((𝑖(+g‘𝑅)𝑋) = (0g‘𝑅) ↔ (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅)))
14	13	elrab 3648	. . 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 2147  ∃!wreu 3367  {crab 3416  ‘cfv 6566  ℩crio 7407  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  0_gc0g 17660  inv_gcminusg 19195

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-pow 5334  ax-pr 5398  ax-un 7783

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-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-pw 4562  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-fv 6574  df-riota 7408  df-ov 7454  df-minusg 19198

This theorem is used by:  primrootsunit1  43413