Theorem smupp1 16450

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 12835	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		seqp1 13981	. . . 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 17	. . 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 6859	. . 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 6859	. . . 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 7397	. . 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 2789	. 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 12458	. . . . . . 7 ⊢ 1 ∈ ℕ0
12	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0)
13	1, 12	nn0addcld 12507	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
14		eqeq1 2733	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15		oveq1 7394	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
16	14, 15	ifbieq2d 4515	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17		eqid 2729	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18		0ex 5262	. . . . . . 7 ⊢ ∅ ∈ V
19		ovex 7420	. . . . . . 7 ⊢ ((𝑁 + 1) − 1) ∈ V
20	18, 19	ifex 4539	. . . . . 6 ⊢ if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
21	16, 17, 20	fvmpt 6968	. . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
22	13, 21	syl 17	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
23		nn0p1nn 12481	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
24	1, 23	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
25	24	nnne0d 12236	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
26		ifnefalse 4500	. . . . 5 ⊢ ((𝑁 + 1) ≠ 0 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
27	25, 26	syl 17	. . . 4 ⊢ (𝜑 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
28	1	nn0cnd 12505	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
29	12	nn0cnd 12505	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
30	28, 29	pncand 11534	. . . 4 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
31	22, 27, 30	3eqtrd 2768	. . 3 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
32	31	oveq2d 7403	. 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 16448	. . . 4 ⊢ (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)
36	35, 1	ffvelcdmd 7057	. . 3 ⊢ (𝜑 → (𝑃‘𝑁) ∈ 𝒫 ℕ0)
37		simpl 482	. . . . 5 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝑃‘𝑁))
38		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
39	38	eleq1d 2813	. . . . . . . 8 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
40	38	oveq2d 7403	. . . . . . . . 9 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → (𝑘 − 𝑦) = (𝑘 − 𝑁))
41	40	eleq1d 2813	. . . . . . . 8 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → ((𝑘 − 𝑦) ∈ 𝐵 ↔ (𝑘 − 𝑁) ∈ 𝐵))
42	39, 41	anbi12d 632	. . . . . . 7 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → ((𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵) ↔ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)))
43	42	rabbidv 3413	. . . . . 6 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)})
44		oveq1 7394	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑁) = (𝑛 − 𝑁))
45	44	eleq1d 2813	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝑘 − 𝑁) ∈ 𝐵 ↔ (𝑛 − 𝑁) ∈ 𝐵))
46	45	anbi2d 630	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵) ↔ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)))
47	46	cbvrabv 3416	. . . . . 6 ⊢ {𝑘 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}
48	43, 47	eqtrdi 2780	. . . . 5 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)})
49	37, 48	oveq12d 7405	. . . 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 7394	. . . . 5 ⊢ (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))
51		eleq1w 2811	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
52		oveq2 7395	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → (𝑛 − 𝑚) = (𝑛 − 𝑦))
53	52	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ((𝑛 − 𝑚) ∈ 𝐵 ↔ (𝑛 − 𝑦) ∈ 𝐵))
54	51, 53	anbi12d 632	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → ((𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)))
55	54	rabbidv 3413	. . . . . . 7 ⊢ (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)})
56		oveq1 7394	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑦) = (𝑛 − 𝑦))
57	56	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((𝑘 − 𝑦) ∈ 𝐵 ↔ (𝑛 − 𝑦) ∈ 𝐵))
58	57	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)))
59	58	cbvrabv 3416	. . . . . . 7 ⊢ {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)}
60	55, 59	eqtr4di 2782	. . . . . 6 ⊢ (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)})
61	60	oveq2d 7403	. . . . 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 7484	. . . 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 7420	. . . 4 ⊢ ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}) ∈ V
64	49, 62, 63	ovmpoa 7544	. . 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 584	. 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 2768	1 ⊢ (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3405   ⊆ wss 3914  ∅c0 4296  ifcif 4488  𝒫 cpw 4563   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  0cc0 11068  1c1 11069   + caddc 11071   − cmin 11405  ℕcn 12186  ℕ₀cn0 12442  ℤ_≥cuz 12793  seqcseq 13966   sadd csad 16390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-had 1594  df-cad 1607  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-seq 13967  df-sad 16421

This theorem is referenced by:  smuval2  16452  smupvallem  16453  smu01lem  16455  smupval  16458  smup1  16459  smueqlem  16460