iswomni0 - Mathbox for Jim Kingdon

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Theorem iswomni0 17075

Description: Weak omniscience stated in terms of equality with 0. Like iswomninn 17074 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)

**Assertion**
Ref	Expression
iswomni0	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))

Distinct variable groups: 𝐴,𝑓,𝑥 𝑓,𝑉,𝑥

Proof of Theorem iswomni0 Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iswomninn 17074	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1))
2		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) = 0)
3	2	oveq2d 6095	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 0) → (1 − (𝑓‘𝑧)) = (1 − 0))
4		1m0e1 9400	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
5	3, 4	eqtrdi 2287	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 0) → (1 − (𝑓‘𝑧)) = 1)
6		1ex 8315	. . . . . . . . . . 11 ⊢ 1 ∈ V
7	6	prid2 3817	. . . . . . . . . 10 ⊢ 1 ∈ {0, 1}
8	5, 7	eqeltrdi 2329	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 0) → (1 − (𝑓‘𝑧)) ∈ {0, 1})
9		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 1) → (𝑓‘𝑧) = 1)
10	9	oveq2d 6095	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 1) → (1 − (𝑓‘𝑧)) = (1 − 1))
11		1m1e0 9356	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
12	10, 11	eqtrdi 2287	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 1) → (1 − (𝑓‘𝑧)) = 0)
13		c0ex 8314	. . . . . . . . . . 11 ⊢ 0 ∈ V
14	13	prid1 3816	. . . . . . . . . 10 ⊢ 0 ∈ {0, 1}
15	12, 14	eqeltrdi 2329	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 1) → (1 − (𝑓‘𝑧)) ∈ {0, 1})
16		elmapi 6938	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ({0, 1} ↑_𝑚 𝐴) → 𝑓:𝐴⟶{0, 1})
17	16	ad2antlr 493	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → 𝑓:𝐴⟶{0, 1})
18		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
19	17, 18	ffvelcdmd 5838	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) ∈ {0, 1})
20		elpri 3731	. . . . . . . . . 10 ⊢ ((𝑓‘𝑧) ∈ {0, 1} → ((𝑓‘𝑧) = 0 ∨ (𝑓‘𝑧) = 1))
21	19, 20	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 0 ∨ (𝑓‘𝑧) = 1))
22	8, 15, 21	mpjaodan 810	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (1 − (𝑓‘𝑧)) ∈ {0, 1})
23	22	fmpttd 5857	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))):𝐴⟶{0, 1})
24		0nn0 9561	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
25		1nn0 9562	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
26		prexg 4347	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ∈ V)
27	24, 25, 26	mp2an 430	. . . . . . . . 9 ⊢ {0, 1} ∈ V
28	27	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → {0, 1} ∈ V)
29		simpl 109	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → 𝐴 ∈ 𝑉)
30	28, 29	elmapd 6930	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) ∈ ({0, 1} ↑_𝑚 𝐴) ↔ (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))):𝐴⟶{0, 1}))
31	23, 30	mpbird 167	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) ∈ ({0, 1} ↑_𝑚 𝐴))
32		fveq1 5692	. . . . . . . . . 10 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) → (𝑔‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥))
33	32	eqeq1d 2247	. . . . . . . . 9 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) → ((𝑔‘𝑥) = 1 ↔ ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1))
34	33	ralbidv 2550	. . . . . . . 8 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 ↔ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1))
35	34	dcbid 850	. . . . . . 7 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) → (DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 ↔ DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1))
36	35	rspcv 2925	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) ∈ ({0, 1} ↑_𝑚 𝐴) → (∀𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 → DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1))
37	31, 36	syl 14	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → (∀𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 → DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1))
38		eqid 2238	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))
39		fveq2 5693	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑓‘𝑧) = (𝑓‘𝑥))
40	39	oveq2d 6095	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (1 − (𝑓‘𝑧)) = (1 − (𝑓‘𝑥)))
41		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
42	22	ralrimiva 2623	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → ∀𝑧 ∈ 𝐴 (1 − (𝑓‘𝑧)) ∈ {0, 1})
43	40	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ((1 − (𝑓‘𝑧)) ∈ {0, 1} ↔ (1 − (𝑓‘𝑥)) ∈ {0, 1}))
44	43	cbvralv 2786	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝐴 (1 − (𝑓‘𝑧)) ∈ {0, 1} ↔ ∀𝑥 ∈ 𝐴 (1 − (𝑓‘𝑥)) ∈ {0, 1})
45	42, 44	sylib 122	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → ∀𝑥 ∈ 𝐴 (1 − (𝑓‘𝑥)) ∈ {0, 1})
46	45	r19.21bi 2638	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (1 − (𝑓‘𝑥)) ∈ {0, 1})
47	38, 40, 41, 46	fvmptd3 5796	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = (1 − (𝑓‘𝑥)))
48	47	eqeq1d 2247	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1 ↔ (1 − (𝑓‘𝑥)) = 1))
49		1cnd 8336	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℂ)
50		0z 9638	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
51		1z 9653	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
52		prssi 3871	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ)
53	50, 51, 52	mp2an 430	. . . . . . . . . . . 12 ⊢ {0, 1} ⊆ ℤ
54	16	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → 𝑓:𝐴⟶{0, 1})
55	54	ffvelcdmda 5837	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ {0, 1})
56	53, 55	sselid 3246	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ℤ)
57	56	zcnd 9752	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ℂ)
58		subsub23 8525	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝑓‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 − (𝑓‘𝑥)) = 1 ↔ (1 − 1) = (𝑓‘𝑥)))
59	49, 57, 49, 58	syl3anc 1278	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((1 − (𝑓‘𝑥)) = 1 ↔ (1 − 1) = (𝑓‘𝑥)))
60	48, 59	bitrd 188	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1 ↔ (1 − 1) = (𝑓‘𝑥)))
61	11	eqeq1i 2246	. . . . . . . . 9 ⊢ ((1 − 1) = (𝑓‘𝑥) ↔ 0 = (𝑓‘𝑥))
62		eqcom 2240	. . . . . . . . 9 ⊢ (0 = (𝑓‘𝑥) ↔ (𝑓‘𝑥) = 0)
63	61, 62	bitri 184	. . . . . . . 8 ⊢ ((1 − 1) = (𝑓‘𝑥) ↔ (𝑓‘𝑥) = 0)
64	60, 63	bitrdi 196	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1 ↔ (𝑓‘𝑥) = 0))
65	64	ralbidva 2546	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → (∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
66	65	dcbid 850	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → (DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1 ↔ DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
67	37, 66	sylibd 149	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → (∀𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
68	67	ralrimdva 2630	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 → ∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
69		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 0) → (𝑔‘𝑧) = 0)
70	69	oveq2d 6095	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 0) → (1 − (𝑔‘𝑧)) = (1 − 0))
71	70, 4	eqtrdi 2287	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 0) → (1 − (𝑔‘𝑧)) = 1)
72	71, 7	eqeltrdi 2329	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 0) → (1 − (𝑔‘𝑧)) ∈ {0, 1})
73		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 1) → (𝑔‘𝑧) = 1)
74	73	oveq2d 6095	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 1) → (1 − (𝑔‘𝑧)) = (1 − 1))
75	74, 11	eqtrdi 2287	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 1) → (1 − (𝑔‘𝑧)) = 0)
76	75, 14	eqeltrdi 2329	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 1) → (1 − (𝑔‘𝑧)) ∈ {0, 1})
77		elmapi 6938	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ ({0, 1} ↑_𝑚 𝐴) → 𝑔:𝐴⟶{0, 1})
78	77	adantl 277	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → 𝑔:𝐴⟶{0, 1})
79	78	ffvelcdmda 5837	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑧) ∈ {0, 1})
80		elpri 3731	. . . . . . . . . 10 ⊢ ((𝑔‘𝑧) ∈ {0, 1} → ((𝑔‘𝑧) = 0 ∨ (𝑔‘𝑧) = 1))
81	79, 80	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑔‘𝑧) = 0 ∨ (𝑔‘𝑧) = 1))
82	72, 76, 81	mpjaodan 810	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (1 − (𝑔‘𝑧)) ∈ {0, 1})
83	82	fmpttd 5857	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))):𝐴⟶{0, 1})
84	27	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → {0, 1} ∈ V)
85		simpl 109	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → 𝐴 ∈ 𝑉)
86	84, 85	elmapd 6930	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) ∈ ({0, 1} ↑_𝑚 𝐴) ↔ (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))):𝐴⟶{0, 1}))
87	83, 86	mpbird 167	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) ∈ ({0, 1} ↑_𝑚 𝐴))
88		fveq1 5692	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) → (𝑓‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥))
89	88	eqeq1d 2247	. . . . . . . . 9 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) → ((𝑓‘𝑥) = 0 ↔ ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0))
90	89	ralbidv 2550	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0))
91	90	dcbid 850	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ↔ DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0))
92	91	rspcv 2925	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) ∈ ({0, 1} ↑_𝑚 𝐴) → (∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 → DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0))
93	87, 92	syl 14	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → (∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 → DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0))
94		eqid 2238	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))
95		fveq2 5693	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑔‘𝑧) = (𝑔‘𝑥))
96	95	oveq2d 6095	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (1 − (𝑔‘𝑧)) = (1 − (𝑔‘𝑥)))
97		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
98	82	ralrimiva 2623	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → ∀𝑧 ∈ 𝐴 (1 − (𝑔‘𝑧)) ∈ {0, 1})
99	96	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ((1 − (𝑔‘𝑧)) ∈ {0, 1} ↔ (1 − (𝑔‘𝑥)) ∈ {0, 1}))
100	99	cbvralv 2786	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝐴 (1 − (𝑔‘𝑧)) ∈ {0, 1} ↔ ∀𝑥 ∈ 𝐴 (1 − (𝑔‘𝑥)) ∈ {0, 1})
101	98, 100	sylib 122	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → ∀𝑥 ∈ 𝐴 (1 − (𝑔‘𝑥)) ∈ {0, 1})
102	101	r19.21bi 2638	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (1 − (𝑔‘𝑥)) ∈ {0, 1})
103	94, 96, 97, 102	fvmptd3 5796	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = (1 − (𝑔‘𝑥)))
104	103	eqeq1d 2247	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0 ↔ (1 − (𝑔‘𝑥)) = 0))
105		1cnd 8336	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℂ)
106	78	ffvelcdmda 5837	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ {0, 1})
107	53, 106	sselid 3246	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ℤ)
108	107	zcnd 9752	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ℂ)
109		0cnd 8313	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℂ)
110		subsub23 8525	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝑔‘𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → ((1 − (𝑔‘𝑥)) = 0 ↔ (1 − 0) = (𝑔‘𝑥)))
111	105, 108, 109, 110	syl3anc 1278	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((1 − (𝑔‘𝑥)) = 0 ↔ (1 − 0) = (𝑔‘𝑥)))
112	104, 111	bitrd 188	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0 ↔ (1 − 0) = (𝑔‘𝑥)))
113	4	eqeq1i 2246	. . . . . . . . 9 ⊢ ((1 − 0) = (𝑔‘𝑥) ↔ 1 = (𝑔‘𝑥))
114		eqcom 2240	. . . . . . . . 9 ⊢ (1 = (𝑔‘𝑥) ↔ (𝑔‘𝑥) = 1)
115	113, 114	bitri 184	. . . . . . . 8 ⊢ ((1 − 0) = (𝑔‘𝑥) ↔ (𝑔‘𝑥) = 1)
116	112, 115	bitrdi 196	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0 ↔ (𝑔‘𝑥) = 1))
117	116	ralbidva 2546	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → (∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1))
118	117	dcbid 850	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → (DECID ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0 ↔ DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1))
119	93, 118	sylibd 149	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → (∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 → DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1))
120	119	ralrimdva 2630	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 → ∀𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1))
121	68, 120	impbid 129	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 ↔ ∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))
122	1, 121	bitrd 188	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ∀wral 2528 Vcvv 2821 ⊆ wss 3220 {cpr 3709 ↦ cmpt 4190 ⟶wf 5371 ‘cfv 5375 (class class class)co 6079 ↑_𝑚 cmap 6916 WOmnicwomni 7497 ℂcc 8171 0cc0 8173 1c1 8174 − cmin 8491 ℕ₀cn0 9546 ℤcz 9627

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-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-addass 8275 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-ltadd 8289

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-recs 6570 df-frec 6656 df-1o 6681 df-2o 6682 df-map 6918 df-womni 7498 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-inn 9288 df-n0 9547 df-z 9628 df-uz 9905

This theorem is referenced by: nconstwlpo 17090

W3C validator