carageniuncl - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > carageniuncl		Structured version Visualization version GIF version

Theorem carageniuncl 45239

Description: The Caratheodory's construction is closed under indexed countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
carageniuncl.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
carageniuncl.s	⊢ 𝑆 = (CaraGen‘𝑂)
carageniuncl.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
carageniuncl.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
carageniuncl.e	⊢ (𝜑 → 𝐸:𝑍⟶𝑆)

**Assertion**
Ref	Expression
carageniuncl	⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ 𝑆)

Distinct variable groups: 𝑛,𝐸 𝑛,𝑀 𝑛,𝑂 𝑛,𝑍 𝜑,𝑛 Allowed substitution hint: 𝑆(𝑛)

Proof of Theorem carageniuncl Dummy variables 𝑎 𝑖 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		carageniuncl.o	. 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		eqid 2733	. 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂
3		carageniuncl.s	. 2 ⊢ 𝑆 = (CaraGen‘𝑂)
4		carageniuncl.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸:𝑍⟶𝑆)
5	4	ffvelcdmda 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝑆)
6		elssuni 4942	. . . . . . 7 ⊢ ((𝐸‘𝑛) ∈ 𝑆 → (𝐸‘𝑛) ⊆ ∪ 𝑆)
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ ∪ 𝑆)
8	1, 3	caragenuni 45227	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom 𝑂)
9	8	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑆 = ∪ dom 𝑂)
10	7, 9	sseqtrd 4023	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
11	10	ralrimiva 3147	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
12		iunss 5049	. . . 4 ⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ ∪ dom 𝑂 ↔ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
13	11, 12	sylibr 233	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
14		carageniuncl.z	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
15	14	fvexi 6906	. . . . . 6 ⊢ 𝑍 ∈ V
16		fvex 6905	. . . . . 6 ⊢ (𝐸‘𝑛) ∈ V
17	15, 16	iunex 7955	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ V
18	17	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ V)
19		elpwg 4606	. . . 4 ⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ V → (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ 𝒫 ∪ dom 𝑂 ↔ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ ∪ dom 𝑂))
20	18, 19	syl 17	. . 3 ⊢ (𝜑 → (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ 𝒫 ∪ dom 𝑂 ↔ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ ∪ dom 𝑂))
21	13, 20	mpbird 257	. 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ 𝒫 ∪ dom 𝑂)
22		iccssxr 13407	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
23	1	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas)
24		elpwi 4610	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂)
25		ssinss1 4238	. . . . . . . 8 ⊢ (𝑎 ⊆ ∪ dom 𝑂 → (𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ ∪ dom 𝑂)
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → (𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ ∪ dom 𝑂)
27	26	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ ∪ dom 𝑂)
28	23, 2, 27	omecl 45219	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ (0[,]+∞))
29	22, 28	sselid 3981	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ^*)
30	24	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂)
31	30	ssdifssd 4143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ⊆ ∪ dom 𝑂)
32	23, 2, 31	omecl 45219	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ (0[,]+∞))
33	22, 32	sselid 3981	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) ∈ ℝ^*)
34	29, 33	xaddcld 13280	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ∈ ℝ^*)
35	23, 2, 30	omecl 45219	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ (0[,]+∞))
36	22, 35	sselid 3981	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ ℝ^*)
37		pnfge 13110	. . . . . . 7 ⊢ (((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ∈ ℝ^* → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ +∞)
38	34, 37	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ +∞)
39	38	adantr 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) = +∞) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ +∞)
40		id 22	. . . . . . 7 ⊢ ((𝑂‘𝑎) = +∞ → (𝑂‘𝑎) = +∞)
41	40	eqcomd 2739	. . . . . 6 ⊢ ((𝑂‘𝑎) = +∞ → +∞ = (𝑂‘𝑎))
42	41	adantl 483	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) = +∞) → +∞ = (𝑂‘𝑎))
43	39, 42	breqtrd 5175	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) = +∞) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ (𝑂‘𝑎))
44		simpl 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → (𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂))
45		rge0ssre 13433	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
46		0xr 11261	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
47	46	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → 0 ∈ ℝ^*)
48		pnfxr 11268	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
49	48	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → +∞ ∈ ℝ^*)
50	44, 35	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → (𝑂‘𝑎) ∈ (0[,]+∞))
51	40	necon3bi 2968	. . . . . . . 8 ⊢ (¬ (𝑂‘𝑎) = +∞ → (𝑂‘𝑎) ≠ +∞)
52	51	adantl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → (𝑂‘𝑎) ≠ +∞)
53	47, 49, 50, 52	eliccelicod 44243	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → (𝑂‘𝑎) ∈ (0[,)+∞))
54	45, 53	sselid 3981	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → (𝑂‘𝑎) ∈ ℝ)
55	23	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑂 ∈ OutMeas)
56	30	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑎 ⊆ ∪ dom 𝑂)
57		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) → (𝑂‘𝑎) ∈ ℝ)
58	57	adantr 482	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑂‘𝑎) ∈ ℝ)
59		carageniuncl.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60	59	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑀 ∈ ℤ)
61	4	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐸:𝑍⟶𝑆)
62		simpr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
63		eqid 2733	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) = (𝑛 ∈ 𝑍 ↦ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))
64		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐸‘𝑚) = (𝐸‘𝑛))
65		oveq2 7417	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑀..^𝑚) = (𝑀..^𝑛))
66	65	iuneq1d 5025	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ∪ 𝑖 ∈ (𝑀..^𝑚)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖))
67	64, 66	difeq12d 4124	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐸‘𝑚) ∖ ∪ 𝑖 ∈ (𝑀..^𝑚)(𝐸‘𝑖)) = ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)))
68	67	cbvmptv 5262	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝑚) ∖ ∪ 𝑖 ∈ (𝑀..^𝑚)(𝐸‘𝑖))) = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑀..^𝑛)(𝐸‘𝑖)))
69	55, 3, 2, 56, 58, 60, 14, 61, 62, 63, 68	carageniuncllem2 45238	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝑎) + 𝑥))
70	69	ralrimiva 3147	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝑎) + 𝑥))
71	34	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ∈ ℝ^*)
72		xralrple 13184	. . . . . . 7 ⊢ ((((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ∈ ℝ^* ∧ (𝑂‘𝑎) ∈ ℝ) → (((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ (𝑂‘𝑎) ↔ ∀𝑥 ∈ ℝ⁺ ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝑎) + 𝑥)))
73	71, 57, 72	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) → (((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ (𝑂‘𝑎) ↔ ∀𝑥 ∈ ℝ⁺ ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ ((𝑂‘𝑎) + 𝑥)))
74	70, 73	mpbird 257	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ (𝑂‘𝑎) ∈ ℝ) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ (𝑂‘𝑎))
75	44, 54, 74	syl2anc 585	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) ∧ ¬ (𝑂‘𝑎) = +∞) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ (𝑂‘𝑎))
76	43, 75	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) ≤ (𝑂‘𝑎))
77	23, 2, 30	omelesplit 45234	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ≤ ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))))
78	34, 36, 76, 77	xrletrid 13134	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) +_𝑒 (𝑂‘(𝑎 ∖ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))) = (𝑂‘𝑎))
79	1, 2, 3, 21, 78	carageneld 45218	1 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4603 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 + caddc 11113 +∞cpnf 11245 ℝ^*cxr 11247 ≤ cle 11249 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 +_𝑒 cxad 13090 [,)cico 13326 [,]cicc 13327 ...cfz 13484 ..^cfzo 13627 OutMeascome 45205 CaraGenccaragen 45207

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xadd 13093 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-sumge0 45079 df-ome 45206 df-caragen 45208

This theorem is referenced by: caragenunicl 45240

W3C validator