Theorem smupp1 16407

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 12789	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2846	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		seqp1 13939	. . . 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 6835	. . 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 6835	. . . 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 7368	. . 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 2796	. 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 12417	. . . . . . 7 ⊢ 1 ∈ ℕ0
12	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0)
13	1, 12	nn0addcld 12466	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
14		eqeq1 2740	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15		oveq1 7365	. . . . . . 7 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
16	14, 15	ifbieq2d 4506	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17		eqid 2736	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18		0ex 5252	. . . . . . 7 ⊢ ∅ ∈ V
19		ovex 7391	. . . . . . 7 ⊢ ((𝑁 + 1) − 1) ∈ V
20	18, 19	ifex 4530	. . . . . 6 ⊢ if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
21	16, 17, 20	fvmpt 6941	. . . . 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 12440	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
24	1, 23	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
25	24	nnne0d 12195	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
26		ifnefalse 4491	. . . . 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 12464	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
29	12	nn0cnd 12464	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
30	28, 29	pncand 11493	. . . 4 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
31	22, 27, 30	3eqtrd 2775	. . 3 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
32	31	oveq2d 7374	. 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 16405	. . . 4 ⊢ (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)
36	35, 1	ffvelcdmd 7030	. . 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 2821	. . . . . . . 8 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
40	38	oveq2d 7374	. . . . . . . . 9 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → (𝑘 − 𝑦) = (𝑘 − 𝑁))
41	40	eleq1d 2821	. . . . . . . 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 3406	. . . . . 6 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)})
44		oveq1 7365	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑁) = (𝑛 − 𝑁))
45	44	eleq1d 2821	. . . . . . . 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 3409	. . . . . 6 ⊢ {𝑘 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑘 − 𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}
48	43, 47	eqtrdi 2787	. . . . 5 ⊢ ((𝑥 = (𝑃‘𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)})
49	37, 48	oveq12d 7376	. . . 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 7365	. . . . 5 ⊢ (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))
51		eleq1w 2819	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
52		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → (𝑛 − 𝑚) = (𝑛 − 𝑦))
53	52	eleq1d 2821	. . . . . . . . 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 3406	. . . . . . 7 ⊢ (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)})
56		oveq1 7365	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑦) = (𝑛 − 𝑦))
57	56	eleq1d 2821	. . . . . . . . 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 3409	. . . . . . 7 ⊢ {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 − 𝑦) ∈ 𝐵)}
60	55, 59	eqtr4di 2789	. . . . . 6 ⊢ (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑘 − 𝑦) ∈ 𝐵)})
61	60	oveq2d 7374	. . . . 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 7453	. . . 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 7391	. . . 4 ⊢ ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}) ∈ V
64	49, 62, 63	ovmpoa 7513	. . 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 2775	1 ⊢ (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃‘𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 − 𝑁) ∈ 𝐵)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  {crab 3399   ⊆ wss 3901  ∅c0 4285  ifcif 4479  𝒫 cpw 4554   ↦ cmpt 5179  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  0cc0 11026  1c1 11027   + caddc 11029   − cmin 11364  ℕcn 12145  ℕ₀cn0 12401  ℤ_≥cuz 12751  seqcseq 13924   sadd csad 16347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-had 1595  df-cad 1608  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-seq 13925  df-sad 16378

This theorem is referenced by:  smuval2  16409  smupvallem  16410  smu01lem  16412  smupval  16415  smup1  16416  smueqlem  16417