Theorem smupp1 16391

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 12777	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		seqp1 13923	. . . 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 6823	. . 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 6823	. . . 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 7359	. . 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 12400	. . . . . . 7 ⊢ 1 ∈ ℕ0
12	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0)
13	1, 12	nn0addcld 12449	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
14		eqeq1 2733	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15		oveq1 7356	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
16	14, 15	ifbieq2d 4503	. . . . . 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 5246	. . . . . . 7 ⊢ ∅ ∈ V
19		ovex 7382	. . . . . . 7 ⊢ ((𝑁 + 1) − 1) ∈ V
20	18, 19	ifex 4527	. . . . . 6 ⊢ if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
21	16, 17, 20	fvmpt 6930	. . . . 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 12423	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
24	1, 23	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
25	24	nnne0d 12178	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
26		ifnefalse 4488	. . . . 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 12447	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
29	12	nn0cnd 12447	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
30	28, 29	pncand 11476	. . . 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 7365	. 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 16389	. . . 4 ⊢ (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)
36	35, 1	ffvelcdmd 7019	. . 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 7365	. . . . . . . . 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 3402	. . . . . 6 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)})
44		oveq1 7356	. . . . . . . . 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 3405	. . . . . 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 7367	. . . 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 7356	. . . . 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 7357	. . . . . . . . . 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 3402	. . . . . . 7 ⊢ (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)})
56		oveq1 7356	. . . . . . . . . 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 3405	. . . . . . 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 7365	. . . . 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 7444	. . . 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 7382	. . . 4 ⊢ ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}) ∈ V
64	49, 62, 63	ovmpoa 7504	. . 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 3394   ⊆ wss 3903  ∅c0 4284  ifcif 4476  𝒫 cpw 4551   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  0cc0 11009  1c1 11010   + caddc 11012   − cmin 11347  ℕcn 12128  ℕ₀cn0 12384  ℤ_≥cuz 12735  seqcseq 13908   sadd csad 16331

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 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-seq 13909  df-sad 16362

This theorem is referenced by:  smuval2  16393  smupvallem  16394  smu01lem  16396  smupval  16399  smup1  16400  smueqlem  16401