Theorem linvh 42092

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 2737	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2737	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2737	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2737	. . . . 5 ⊢ (invg‘𝑅) = (invg‘𝑅)
6	2, 3, 4, 5	grpinvval 19020	. . . 4 ⊢ (𝑋 ∈ (Base‘𝑅) → ((invg‘𝑅)‘𝑋) = (℩𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)))
7	1, 6	syl 17	. . 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 7411	. . . 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 17	. . 3 ⊢ (𝜑 → (℩𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
11	7, 10	eqeltrd 2841	. 2 ⊢ (𝜑 → ((invg‘𝑅)‘𝑋) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
12		oveq1 7445	. . . . 5 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → (𝑖(+g‘𝑅)𝑋) = (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋))
13	12	eqeq1d 2739	. . . 4 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → ((𝑖(+g‘𝑅)𝑋) = (0g‘𝑅) ↔ (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅)))
14	13	elrab 3698	. . 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 496	. 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 17	1 ⊢ (𝜑 → (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∃!wreu 3378  {crab 3436  ‘cfv 6569  ℩crio 7394  (class class class)co 7438  Basecbs 17254  +_gcplusg 17307  0_gc0g 17495  inv_gcminusg 18974

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fv 6577  df-riota 7395  df-ov 7441  df-minusg 18977

This theorem is referenced by:  primrootsunit1  42093