Theorem idvalriota 18867

Description: The unique value of the group identity element. (Contributed by FL, 12-Dec-2009.) (Revised by AV, 24-Aug-2026.)

Hypotheses

Ref	Expression
grpidval.b	⊢ 𝐵 = (Base‘𝐺)
grpidval.p	⊢ + = (+g‘𝐺)
grpidval.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
idvalriota	⊢ 0 = (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

Distinct variable groups:   𝑥,𝑒,𝐵   𝑒,𝐺,𝑥

Allowed substitution hints:   + (𝑥, 𝑒)   0 (𝑥, 𝑒)

Proof of Theorem idvalriota

Step	Hyp	Ref	Expression
1		grpidval.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		grpidval.p	. . 3 ⊢ + = (+g‘𝐺)
3		grpidval.o	. . 3 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grpidval 18866	. 2 ⊢ 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
5		df-riota 7397	. 2 ⊢ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
6	4, 5	eqtr4i 2789	1 ⊢ 0 = (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ℩cio 6507  ‘cfv 6555  ℩crio 7396  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  0_gc0g 17636

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 5252  ax-nul 5263  ax-pr 5395

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-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563  df-riota 7397  df-ov 7443  df-0g 17638

This theorem is used by: (None)