Theorem 0g0 17086

Description: The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)

Assertion

Ref	Expression
0g0	⊢ ∅ = (0g‘∅)

Proof of Theorem 0g0

Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		base0 15740	. . 3 ⊢ ∅ = (Base‘∅)
2		eqid 2610	. . 3 ⊢ (+g‘∅) = (+g‘∅)
3		eqid 2610	. . 3 ⊢ (0g‘∅) = (0g‘∅)
4	1, 2, 3	grpidval 17083	. 2 ⊢ (0g‘∅) = (℩𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)))
5		noel 3878	. . . . . 6 ⊢ ¬ 𝑒 ∈ ∅
6	5	intnanr 952	. . . . 5 ⊢ ¬ (𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))
7	6	nex 1722	. . . 4 ⊢ ¬ ∃𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))
8		euex 2482	. . . 4 ⊢ (∃!𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)) → ∃𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)))
9	7, 8	mto 187	. . 3 ⊢ ¬ ∃!𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))
10		iotanul 5783	. . 3 ⊢ (¬ ∃!𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)) → (℩𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))) = ∅)
11	9, 10	ax-mp 5	. 2 ⊢ (℩𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))) = ∅
12	4, 11	eqtr2i 2633	1 ⊢ ∅ = (0g‘∅)

Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  ∅c0 3874  ℩cio 5766  ‘cfv 5804  (class class class)co 6549  +_gcplusg 15768  0_gc0g 15923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-slot 15699  df-base 15700  df-0g 15925

This theorem is referenced by:  frmd0  17220  ringidval  18326