Theorem linvh 42350

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 2736	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2736	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2736	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		eqid 2736	. . . . 5 ⊢ (invg‘𝑅) = (invg‘𝑅)
6	2, 3, 4, 5	grpinvval 18910	. . . 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 7331	. . . 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 2836	. 2 ⊢ (𝜑 → ((invg‘𝑅)‘𝑋) ∈ {𝑖 ∈ (Base‘𝑅) ∣ (𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)})
12		oveq1 7365	. . . . 5 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → (𝑖(+g‘𝑅)𝑋) = (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋))
13	12	eqeq1d 2738	. . . 4 ⊢ (𝑖 = ((invg‘𝑅)‘𝑋) → ((𝑖(+g‘𝑅)𝑋) = (0g‘𝑅) ↔ (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅)))
14	13	elrab 3646	. . 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 1541   ∈ wcel 2113  ∃!wreu 3348  {crab 3399  ‘cfv 6492  ℩crio 7314  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  0_gc0g 17359  inv_gcminusg 18864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7315  df-ov 7361  df-minusg 18867

This theorem is referenced by:  primrootsunit1  42351