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Theorem grpidvalg 12791
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 12706 . . 3 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
3 fveq2 5515 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 grpidval.b . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2228 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65eleq2d 2247 . . . . 5 (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒𝐵))
7 fveq2 5515 . . . . . . . . . 10 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 grpidval.p . . . . . . . . . 10 + = (+g𝐺)
97, 8eqtr4di 2228 . . . . . . . . 9 (𝑔 = 𝐺 → (+g𝑔) = + )
109oveqd 5891 . . . . . . . 8 (𝑔 = 𝐺 → (𝑒(+g𝑔)𝑥) = (𝑒 + 𝑥))
1110eqeq1d 2186 . . . . . . 7 (𝑔 = 𝐺 → ((𝑒(+g𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
129oveqd 5891 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑒) = (𝑥 + 𝑒))
1312eqeq1d 2186 . . . . . . 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 2684 . . . . 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 5199 . . 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 2748 . . 3 (𝐺𝑉𝐺 ∈ V)
19 df-riota 5830 . . . 4 (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
20 basfn 12519 . . . . . . 7 Base Fn V
21 funfvex 5532 . . . . . . . 8 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
2221funfni 5316 . . . . . . 7 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
2320, 18, 22sylancr 414 . . . . . 6 (𝐺𝑉 → (Base‘𝐺) ∈ V)
244, 23eqeltrid 2264 . . . . 5 (𝐺𝑉𝐵 ∈ V)
25 riotaexg 5834 . . . . 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 2265 . . 3 (𝐺𝑉 → (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V)
282, 17, 18, 27fvmptd3 5609 . 2 (𝐺𝑉 → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
291, 28eqtrid 2222 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 2148  wral 2455  Vcvv 2737  cio 5176   Fn wfn 5211  cfv 5216  crio 5829  (class class class)co 5874  Basecbs 12461  +gcplusg 12535  0gc0g 12704
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-inn 8919  df-ndx 12464  df-slot 12465  df-base 12467  df-0g 12706
This theorem is referenced by:  grpidpropdg  12792  0g0  12794  ismgmid  12795  sgrpidmndm  12820  dfur2g  13143  oppr0g  13249  oppr1g  13250
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