Theorem grpidvalg 13016

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	⊢ + = (+g‘𝐺)
grpidval.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
grpidvalg	⊢ (𝐺 ∈ 𝑉 → 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((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.

Step	Hyp	Ref	Expression
1		grpidval.o	. 2 ⊢ 0 = (0g‘𝐺)
2		df-0g 12929	. . 3 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
3		fveq2 5558	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		grpidval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2247	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	eleq2d 2266	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒 ∈ 𝐵))
7		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		grpidval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2247	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	oveqd 5939	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑒(+g‘𝑔)𝑥) = (𝑒 + 𝑥))
11	10	eqeq1d 2205	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑒(+g‘𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
12	9	oveqd 5939	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑒) = (𝑥 + 𝑒))
13	12	eqeq1d 2205	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑥(+g‘𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
14	11, 13	anbi12d 473	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
15	5, 14	raleqbidv 2709	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
16	6, 15	anbi12d 473	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) ↔ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
17	16	iotabidv 5241	. . 3 ⊢ (𝑔 = 𝐺 → (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
18		elex 2774	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
19		df-riota 5877	. . . 4 ⊢ (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
20		basfn 12736	. . . . . . 7 ⊢ Base Fn V
21		funfvex 5575	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
22	21	funfni 5358	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
23	20, 18, 22	sylancr 414	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
24	4, 23	eqeltrid 2283	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V)
25		riotaexg 5881	. . . . 5 ⊢ (𝐵 ∈ V → (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ V)
26	24, 25	syl 14	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ V)
27	19, 26	eqeltrrid 2284	. . 3 ⊢ (𝐺 ∈ 𝑉 → (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V)
28	2, 17, 18, 27	fvmptd3 5655	. 2 ⊢ (𝐺 ∈ 𝑉 → (0g‘𝐺) = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
29	1, 28	eqtrid 2241	1 ⊢ (𝐺 ∈ 𝑉 → 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ℩cio 5217   Fn wfn 5253  ‘cfv 5258  ℩crio 5876  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-0g 12929

This theorem is referenced by:  grpidpropdg  13017  0g0  13019  ismgmid  13020  sgrpidmndm  13061  dfur2g  13518  oppr0g  13637  oppr1g  13638