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Theorem grpidvalg 12658
Description: The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
grpidval.b 𝐵 = (Base‘𝐺)
grpidval.p + = (+g𝐺)
grpidval.o 0 = (0g𝐺)
Assertion
Ref Expression
grpidvalg (𝐺𝑉0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
Distinct variable groups:   𝑥,𝑒,𝐵   𝑒,𝐺,𝑥
Allowed substitution hints:   + (𝑥,𝑒)   𝑉(𝑥,𝑒)   0 (𝑥,𝑒)

Proof of Theorem grpidvalg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpidval.o . 2 0 = (0g𝐺)
2 df-0g 12629 . . 3 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
3 fveq2 5507 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 grpidval.b . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2226 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65eleq2d 2245 . . . . 5 (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒𝐵))
7 fveq2 5507 . . . . . . . . . 10 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 grpidval.p . . . . . . . . . 10 + = (+g𝐺)
97, 8eqtr4di 2226 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = + )
109oveqd 5882 . . . . . . . 8 (𝑔 = 𝐺 → (𝑒(+g𝑔)𝑥) = (𝑒 + 𝑥))
1110eqeq1d 2184 . . . . . . 7 (𝑔 = 𝐺 → ((𝑒(+g𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
129oveqd 5882 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑒) = (𝑥 + 𝑒))
1312eqeq1d 2184 . . . . . . 7 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
1411, 13anbi12d 473 . . . . . 6 (𝑔 = 𝐺 → (((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
155, 14raleqbidv 2682 . . . . 5 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
166, 15anbi12d 473 . . . 4 (𝑔 = 𝐺 → ((𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥)) ↔ (𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
1716iotabidv 5191 . . 3 (𝑔 = 𝐺 → (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
18 elex 2746 . . 3 (𝐺𝑉𝐺 ∈ V)
19 df-riota 5821 . . . 4 (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
20 basfn 12486 . . . . . . 7 Base Fn V
21 funfvex 5524 . . . . . . . 8 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
2221funfni 5308 . . . . . . 7 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
2320, 18, 22sylancr 414 . . . . . 6 (𝐺𝑉 → (Base‘𝐺) ∈ V)
244, 23eqeltrid 2262 . . . . 5 (𝐺𝑉𝐵 ∈ V)
25 riotaexg 5825 . . . . 5 (𝐵 ∈ V → (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ V)
2624, 25syl 14 . . . 4 (𝐺𝑉 → (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ V)
2719, 26eqeltrrid 2263 . . 3 (𝐺𝑉 → (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V)
282, 17, 18, 27fvmptd3 5601 . 2 (𝐺𝑉 → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
291, 28eqtrid 2220 1 (𝐺𝑉0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2146  wral 2453  Vcvv 2735  cio 5168   Fn wfn 5203  cfv 5208  crio 5820  (class class class)co 5865  Basecbs 12429  +gcplusg 12493  0gc0g 12627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-inn 8893  df-ndx 12432  df-slot 12433  df-base 12435  df-0g 12629
This theorem is referenced by:  grpidpropdg  12659  0g0  12661  ismgmid  12662  sgrpidmndm  12687  dfur2g  12942  oppr0g  13046  oppr1g  13047
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