iswomni0 - Mathbox for Jim Kingdon

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Theorem iswomni0 16064

Description: Weak omniscience stated in terms of equality with 0. Like iswomninn 16063 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 16063	. 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 5967	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 0) → (1 − (𝑓‘𝑧)) = (1 − 0))
4		1m0e1 9156	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
5	3, 4	eqtrdi 2255	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 0) → (1 − (𝑓‘𝑧)) = 1)
6		1ex 8074	. . . . . . . . . . 11 ⊢ 1 ∈ V
7	6	prid2 3741	. . . . . . . . . 10 ⊢ 1 ∈ {0, 1}
8	5, 7	eqeltrdi 2297	. . . . . . . . 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 5967	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 1) → (1 − (𝑓‘𝑧)) = (1 − 1))
11		1m1e0 9112	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
12	10, 11	eqtrdi 2255	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 1) → (1 − (𝑓‘𝑧)) = 0)
13		c0ex 8073	. . . . . . . . . . 11 ⊢ 0 ∈ V
14	13	prid1 3740	. . . . . . . . . 10 ⊢ 0 ∈ {0, 1}
15	12, 14	eqeltrdi 2297	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑓‘𝑧) = 1) → (1 − (𝑓‘𝑧)) ∈ {0, 1})
16		elmapi 6764	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ({0, 1} ↑_𝑚 𝐴) → 𝑓:𝐴⟶{0, 1})
17	16	ad2antlr 489	. . . . . . . . . . 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 5723	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) ∈ {0, 1})
20		elpri 3657	. . . . . . . . . 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 800	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (1 − (𝑓‘𝑧)) ∈ {0, 1})
23	22	fmpttd 5742	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))):𝐴⟶{0, 1})
24		0nn0 9317	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
25		1nn0 9318	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
26		prexg 4259	. . . . . . . . . 10 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ∈ V)
27	24, 25, 26	mp2an 426	. . . . . . . . 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 6756	. . . . . . 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 5582	. . . . . . . . . 10 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) → (𝑔‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥))
33	32	eqeq1d 2215	. . . . . . . . 9 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) → ((𝑔‘𝑥) = 1 ↔ ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1))
34	33	ralbidv 2507	. . . . . . . 8 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1 ↔ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1))
35	34	dcbid 840	. . . . . . 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 2874	. . . . . 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 2206	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧))) = (𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))
39		fveq2 5583	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑓‘𝑧) = (𝑓‘𝑥))
40	39	oveq2d 5967	. . . . . . . . . . 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 2580	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → ∀𝑧 ∈ 𝐴 (1 − (𝑓‘𝑧)) ∈ {0, 1})
43	40	eleq1d 2275	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ((1 − (𝑓‘𝑧)) ∈ {0, 1} ↔ (1 − (𝑓‘𝑥)) ∈ {0, 1}))
44	43	cbvralv 2739	. . . . . . . . . . . . 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 2595	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (1 − (𝑓‘𝑥)) ∈ {0, 1})
47	38, 40, 41, 46	fvmptd3 5680	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = (1 − (𝑓‘𝑥)))
48	47	eqeq1d 2215	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑓‘𝑧)))‘𝑥) = 1 ↔ (1 − (𝑓‘𝑥)) = 1))
49		1cnd 8095	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℂ)
50		0z 9390	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
51		1z 9405	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
52		prssi 3793	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ)
53	50, 51, 52	mp2an 426	. . . . . . . . . . . 12 ⊢ {0, 1} ⊆ ℤ
54	16	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) → 𝑓:𝐴⟶{0, 1})
55	54	ffvelcdmda 5722	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ {0, 1})
56	53, 55	sselid 3192	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ℤ)
57	56	zcnd 9503	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ ℂ)
58		subsub23 8284	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝑓‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → ((1 − (𝑓‘𝑥)) = 1 ↔ (1 − 1) = (𝑓‘𝑥)))
59	49, 57, 49, 58	syl3anc 1250	. . . . . . . . 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 2214	. . . . . . . . 9 ⊢ ((1 − 1) = (𝑓‘𝑥) ↔ 0 = (𝑓‘𝑥))
62		eqcom 2208	. . . . . . . . 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 2503	. . . . . 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 840	. . . . 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 2587	. . 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 5967	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 0) → (1 − (𝑔‘𝑧)) = (1 − 0))
71	70, 4	eqtrdi 2255	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 0) → (1 − (𝑔‘𝑧)) = 1)
72	71, 7	eqeltrdi 2297	. . . . . . . . 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 5967	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 1) → (1 − (𝑔‘𝑧)) = (1 − 1))
75	74, 11	eqtrdi 2255	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 1) → (1 − (𝑔‘𝑧)) = 0)
76	75, 14	eqeltrdi 2297	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑔‘𝑧) = 1) → (1 − (𝑔‘𝑧)) ∈ {0, 1})
77		elmapi 6764	. . . . . . . . . . . 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 5722	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑧) ∈ {0, 1})
80		elpri 3657	. . . . . . . . . 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 800	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (1 − (𝑔‘𝑧)) ∈ {0, 1})
83	82	fmpttd 5742	. . . . . . 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 6756	. . . . . . 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 5582	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) → (𝑓‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥))
89	88	eqeq1d 2215	. . . . . . . . 9 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) → ((𝑓‘𝑥) = 0 ↔ ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0))
90	89	ralbidv 2507	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0))
91	90	dcbid 840	. . . . . . 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 2874	. . . . . 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 2206	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧))) = (𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))
95		fveq2 5583	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑔‘𝑧) = (𝑔‘𝑥))
96	95	oveq2d 5967	. . . . . . . . . . 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 2580	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) → ∀𝑧 ∈ 𝐴 (1 − (𝑔‘𝑧)) ∈ {0, 1})
99	96	eleq1d 2275	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ((1 − (𝑔‘𝑧)) ∈ {0, 1} ↔ (1 − (𝑔‘𝑥)) ∈ {0, 1}))
100	99	cbvralv 2739	. . . . . . . . . . . . 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 2595	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (1 − (𝑔‘𝑥)) ∈ {0, 1})
103	94, 96, 97, 102	fvmptd3 5680	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = (1 − (𝑔‘𝑥)))
104	103	eqeq1d 2215	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1 − (𝑔‘𝑧)))‘𝑥) = 0 ↔ (1 − (𝑔‘𝑥)) = 0))
105		1cnd 8095	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℂ)
106	78	ffvelcdmda 5722	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ {0, 1})
107	53, 106	sselid 3192	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ℤ)
108	107	zcnd 9503	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ℂ)
109		0cnd 8072	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑_𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℂ)
110		subsub23 8284	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝑔‘𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → ((1 − (𝑔‘𝑥)) = 0 ↔ (1 − 0) = (𝑔‘𝑥)))
111	105, 108, 109, 110	syl3anc 1250	. . . . . . . . 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 2214	. . . . . . . . 9 ⊢ ((1 − 0) = (𝑔‘𝑥) ↔ 1 = (𝑔‘𝑥))
114		eqcom 2208	. . . . . . . . 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 2503	. . . . . 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 840	. . . . 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 2587	. . 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 710 DECID wdc 836 = wceq 1373 ∈ wcel 2177 ∀wral 2485 Vcvv 2773 ⊆ wss 3167 {cpr 3635 ↦ cmpt 4109 ⟶wf 5272 ‘cfv 5276 (class class class)co 5951 ↑_𝑚 cmap 6742 WOmnicwomni 7272 ℂcc 7930 0cc0 7932 1c1 7933 − cmin 8250 ℕ₀cn0 9302 ℤcz 9379

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-addcom 8032 ax-addass 8034 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-0id 8040 ax-rnegex 8041 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-ltadd 8048

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-recs 6398 df-frec 6484 df-1o 6509 df-2o 6510 df-map 6744 df-womni 7273 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-inn 9044 df-n0 9303 df-z 9380 df-uz 9656

This theorem is referenced by: nconstwlpo 16079

W3C validator