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Theorem 0g0 17863
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.
StepHypRef Expression
1 base0 16526 . . 3 ∅ = (Base‘∅)
2 eqid 2826 . . 3 (+g‘∅) = (+g‘∅)
3 eqid 2826 . . 3 (0g‘∅) = (0g‘∅)
41, 2, 3grpidval 17860 . 2 (0g‘∅) = (℩𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)))
5 noel 4300 . . . . . 6 ¬ 𝑒 ∈ ∅
65intnanr 488 . . . . 5 ¬ (𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))
76nex 1794 . . . 4 ¬ ∃𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))
8 euex 2660 . . . 4 (∃!𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)) → ∃𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)))
97, 8mto 198 . . 3 ¬ ∃!𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))
10 iotanul 6331 . . 3 (¬ ∃!𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥)) → (℩𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))) = ∅)
119, 10ax-mp 5 . 2 (℩𝑒(𝑒 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ((𝑒(+g‘∅)𝑥) = 𝑥 ∧ (𝑥(+g‘∅)𝑒) = 𝑥))) = ∅
124, 11eqtr2i 2850 1 ∅ = (0g‘∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1530  wex 1773  wcel 2107  ∃!weu 2651  wral 3143  c0 4295  cio 6310  cfv 6352  (class class class)co 7148  +gcplusg 16555  0gc0g 16703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-ov 7151  df-slot 16477  df-base 16479  df-0g 16705
This theorem is referenced by:  frmd0  18008  ringidval  19173
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