Theorem smupp1 16573

Description: The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)

Hypotheses

Ref	Expression
smuval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ0)
smuval.b	⊢ (𝜑 → 𝐵 ⊆ ℕ0)
smuval.p	⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smuval.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
smupp1	⊢ (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}))

Distinct variable groups:   𝑚,𝑛,𝑝,𝐴   𝑛,𝑁   𝜑,𝑛   𝐵,𝑚,𝑛,𝑝

Allowed substitution hints:   𝜑(𝑚, 𝑝)   𝑃(𝑚, 𝑛, 𝑝)   𝑁(𝑚, 𝑝)

Proof of Theorem smupp1

Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smuval.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		nn0uz 12928	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2870	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		seqp1 14083	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
6		smuval.p	. . . 4 ⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
7	6	fveq1i 6880	. . 3 ⊢ (𝑃‘(𝑁 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1))
8	6	fveq1i 6880	. . . 4 ⊢ (𝑃‘𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)
9	8	oveq1i 7424	. . 3 ⊢ ((𝑃‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)))
10	5, 7, 9	3eqtr4g 2820	. 2 ⊢ (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
11		1nn0 12547	. . . . . . 7 ⊢ 1 ∈ ℕ0
12	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0)
13	1, 12	nn0addcld 12596	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
14		eqeq1 2764	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15		oveq1 7421	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
16	14, 15	ifbieq2d 4509	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17		eqid 2760	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18		0ex 5264	. . . . . . 7 ⊢ ∅ ∈ V
19		ovex 7447	. . . . . . 7 ⊢ ((𝑁 + 1) − 1) ∈ V
20	18, 19	ifex 4533	. . . . . 6 ⊢ if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
21	16, 17, 20	fvmpt 6987	. . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
22	13, 21	syl 18	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
23		nn0p1nn 12570	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
24	1, 23	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
25	24	nnne0d 12313	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
26		ifnefalse 4494	. . . . 5 ⊢ ((𝑁 + 1) ≠ 0 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
27	25, 26	syl 18	. . . 4 ⊢ (𝜑 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
28	1	nn0cnd 12594	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
29	12	nn0cnd 12594	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
30	28, 29	pncand 11597	. . . 4 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
31	22, 27, 30	3eqtrd 2799	. . 3 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
32	31	oveq2d 7430	. 2 ⊢ (𝜑 → ((𝑃‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((𝑃‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑁))
33		smuval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℕ0)
34		smuval.b	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℕ0)
35	33, 34, 6	smupf 16571	. . . 4 ⊢ (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)
36	35, 1	ffvelcdmd 7079	. . 3 ⊢ (𝜑 → (𝑃‘𝑁) ∈ 𝒫 ℕ0)
37		simpl 488	. . . . 5 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝑃‘𝑁))
38		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
39	38	eleq1d 2845	. . . . . . . 8 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
40	38	oveq2d 7430	. . . . . . . . 9 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → (𝑘 − 𝑦) = (𝑘 − 𝑁))
41	40	eleq1d 2845	. . . . . . . 8 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → ((𝑘 − 𝑦) ∈ 𝐵 ↔ (𝑘 − 𝑁) ∈ 𝐵))
42	39, 41	anbi12d 644	. . . . . . 7 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → ((𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵) ↔ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)))
43	42	rabbidv 3419	. . . . . 6 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)})
44		oveq1 7421	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑁) = (𝑛 − 𝑁))
45	44	eleq1d 2845	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝑘 − 𝑁) ∈ 𝐵 ↔ (𝑛 − 𝑁) ∈ 𝐵))
46	45	anbi2d 642	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵) ↔ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)))
47	46	cbvrabv 3422	. . . . . 6 ⊢ {𝑘 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}
48	43, 47	eqtrdi 2811	. . . . 5 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)})
49	37, 48	oveq12d 7432	. . . 4 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)}) = ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}))
50		oveq1 7421	. . . . 5 ⊢ (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))
51		eleq1w 2843	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
52		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → (𝑛 − 𝑚) = (𝑛 − 𝑦))
53	52	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ((𝑛 − 𝑚) ∈ 𝐵 ↔ (𝑛 − 𝑦) ∈ 𝐵))
54	51, 53	anbi12d 644	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → ((𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)))
55	54	rabbidv 3419	. . . . . . 7 ⊢ (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)})
56		oveq1 7421	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑦) = (𝑛 − 𝑦))
57	56	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝑘 − 𝑦) ∈ 𝐵 ↔ (𝑛 − 𝑦) ∈ 𝐵))
58	57	anbi2d 642	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)))
59	58	cbvrabv 3422	. . . . . . 7 ⊢ {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)}
60	55, 59	eqtr4di 2813	. . . . . 6 ⊢ (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)})
61	60	oveq2d 7430	. . . . 5 ⊢ (𝑚 = 𝑦 → (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)}))
62	50, 61	cbvmpov 7509	. . . 4 ⊢ (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})) = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ ℕ0 ↦ (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)}))
63		ovex 7447	. . . 4 ⊢ ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}) ∈ V
64	49, 62, 63	ovmpoa 7569	. . 3 ⊢ (((𝑃‘𝑁) ∈ 𝒫 ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝑃‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑁) = ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}))
65	36, 1, 64	syl2anc 596	. 2 ⊢ (𝜑 → ((𝑃‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑁) = ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}))
66	10, 32, 65	3eqtrd 2799	1 ⊢ (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  {crab 3412   ⊆ wss 3899  ∅c0 4279  ifcif 4482  𝒫 cpw 4557   ↦ cmpt 5186  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  0cc0 11127  1c1 11128   + caddc 11130   − cmin 11468  ℕcn 12260  ℕ₀cn0 12531  ℤ_≥cuz 12890  seqcseq 14068   sadd csad 16513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-had 1624  df-cad 1640  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13565  df-seq 14069  df-sad 16544

This theorem is used by:  smuval2  16575  smupvallem  16576  smu01lem  16578  smupval  16581  smup1  16582  smueqlem  16583