MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smupp1 Structured version   Visualization version   GIF version
Theorem smupp1 16417
Description: The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
smuval.a (πœ‘ β†’ 𝐴 βŠ† β„•0)
smuval.b (πœ‘ β†’ 𝐡 βŠ† β„•0)
smuval.p 𝑃 = seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))
smuval.n (πœ‘ β†’ 𝑁 ∈ β„•0)
Assertion
Ref Expression
smupp1 (πœ‘ β†’ (π‘ƒβ€˜(𝑁 + 1)) = ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}))
Distinct variable groups:   π‘š,𝑛,𝑝,𝐴   𝑛,𝑁   πœ‘,𝑛   𝐡,π‘š,𝑛,𝑝
Allowed substitution hints:   πœ‘(π‘š,𝑝)   𝑃(π‘š,𝑛,𝑝)   𝑁(π‘š,𝑝)
Proof of Theorem smupp1
Dummy variables π‘˜ π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smuval.n . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„•0)
2 nn0uz 12860 . . . . 5 β„•0 = (β„€β‰₯β€˜0)
31, 2eleqtrdi 2843 . . . 4 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
4 seqp1 13977 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ (seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))β€˜(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))β€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1))))
53, 4syl 17 . . 3 (πœ‘ β†’ (seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))β€˜(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))β€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1))))
6 smuval.p . . . 4 𝑃 = seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))
76fveq1i 6889 . . 3 (π‘ƒβ€˜(𝑁 + 1)) = (seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))β€˜(𝑁 + 1))
86fveq1i 6889 . . . 4 (π‘ƒβ€˜π‘) = (seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))β€˜π‘)
98oveq1i 7415 . . 3 ((π‘ƒβ€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1))) = ((seq0((𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})), (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))))β€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1)))
105, 7, 93eqtr4g 2797 . 2 (πœ‘ β†’ (π‘ƒβ€˜(𝑁 + 1)) = ((π‘ƒβ€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1))))
11 1nn0 12484 . . . . . . 7 1 ∈ β„•0
1211a1i 11 . . . . . 6 (πœ‘ β†’ 1 ∈ β„•0)
131, 12nn0addcld 12532 . . . . 5 (πœ‘ β†’ (𝑁 + 1) ∈ β„•0)
14 eqeq1 2736 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 7412 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
1614, 15ifbieq2d 4553 . . . . . 6 (𝑛 = (𝑁 + 1) β†’ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)) = if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)))
17 eqid 2732 . . . . . 6 (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))) = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))
18 0ex 5306 . . . . . . 7 βˆ… ∈ V
19 ovex 7438 . . . . . . 7 ((𝑁 + 1) βˆ’ 1) ∈ V
2018, 19ifex 4577 . . . . . 6 if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)) ∈ V
2116, 17, 20fvmpt 6995 . . . . 5 ((𝑁 + 1) ∈ β„•0 β†’ ((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1)) = if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)))
2213, 21syl 17 . . . 4 (πœ‘ β†’ ((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1)) = if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)))
23 nn0p1nn 12507 . . . . . . 7 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
241, 23syl 17 . . . . . 6 (πœ‘ β†’ (𝑁 + 1) ∈ β„•)
2524nnne0d 12258 . . . . 5 (πœ‘ β†’ (𝑁 + 1) β‰  0)
26 ifnefalse 4539 . . . . 5 ((𝑁 + 1) β‰  0 β†’ if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)) = ((𝑁 + 1) βˆ’ 1))
2725, 26syl 17 . . . 4 (πœ‘ β†’ if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)) = ((𝑁 + 1) βˆ’ 1))
281nn0cnd 12530 . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„‚)
2912nn0cnd 12530 . . . . 5 (πœ‘ β†’ 1 ∈ β„‚)
3028, 29pncand 11568 . . . 4 (πœ‘ β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
3122, 27, 303eqtrd 2776 . . 3 (πœ‘ β†’ ((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1)) = 𝑁)
3231oveq2d 7421 . 2 (πœ‘ β†’ ((π‘ƒβ€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1))) = ((π‘ƒβ€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))𝑁))
33 smuval.a . . . . 5 (πœ‘ β†’ 𝐴 βŠ† β„•0)
34 smuval.b . . . . 5 (πœ‘ β†’ 𝐡 βŠ† β„•0)
3533, 34, 6smupf 16415 . . . 4 (πœ‘ β†’ 𝑃:β„•0βŸΆπ’« β„•0)
3635, 1ffvelcdmd 7084 . . 3 (πœ‘ β†’ (π‘ƒβ€˜π‘) ∈ 𝒫 β„•0)
37 simpl 483 . . . . 5 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ π‘₯ = (π‘ƒβ€˜π‘))
38 simpr 485 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ 𝑦 = 𝑁)
3938eleq1d 2818 . . . . . . . 8 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
4038oveq2d 7421 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (π‘˜ βˆ’ 𝑦) = (π‘˜ βˆ’ 𝑁))
4140eleq1d 2818 . . . . . . . 8 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ ((π‘˜ βˆ’ 𝑦) ∈ 𝐡 ↔ (π‘˜ βˆ’ 𝑁) ∈ 𝐡))
4239, 41anbi12d 631 . . . . . . 7 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ ((𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡) ↔ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)))
4342rabbidv 3440 . . . . . 6 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)} = {π‘˜ ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)})
44 oveq1 7412 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑁) = (𝑛 βˆ’ 𝑁))
4544eleq1d 2818 . . . . . . . 8 (π‘˜ = 𝑛 β†’ ((π‘˜ βˆ’ 𝑁) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑁) ∈ 𝐡))
4645anbi2d 629 . . . . . . 7 (π‘˜ = 𝑛 β†’ ((𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡) ↔ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)))
4746cbvrabv 3442 . . . . . 6 {π‘˜ ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}
4843, 47eqtrdi 2788 . . . . 5 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)})
4937, 48oveq12d 7423 . . . 4 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (π‘₯ sadd {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)}) = ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}))
50 oveq1 7412 . . . . 5 (𝑝 = π‘₯ β†’ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}) = (π‘₯ sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))
51 eleq1w 2816 . . . . . . . . 9 (π‘š = 𝑦 β†’ (π‘š ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
52 oveq2 7413 . . . . . . . . . 10 (π‘š = 𝑦 β†’ (𝑛 βˆ’ π‘š) = (𝑛 βˆ’ 𝑦))
5352eleq1d 2818 . . . . . . . . 9 (π‘š = 𝑦 β†’ ((𝑛 βˆ’ π‘š) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑦) ∈ 𝐡))
5451, 53anbi12d 631 . . . . . . . 8 (π‘š = 𝑦 β†’ ((π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)))
5554rabbidv 3440 . . . . . . 7 (π‘š = 𝑦 β†’ {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)})
56 oveq1 7412 . . . . . . . . . 10 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑦) = (𝑛 βˆ’ 𝑦))
5756eleq1d 2818 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ ((π‘˜ βˆ’ 𝑦) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑦) ∈ 𝐡))
5857anbi2d 629 . . . . . . . 8 (π‘˜ = 𝑛 β†’ ((𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)))
5958cbvrabv 3442 . . . . . . 7 {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)}
6055, 59eqtr4di 2790 . . . . . 6 (π‘š = 𝑦 β†’ {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)} = {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)})
6160oveq2d 7421 . . . . 5 (π‘š = 𝑦 β†’ (π‘₯ sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}) = (π‘₯ sadd {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)}))
6250, 61cbvmpov 7500 . . . 4 (𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)})) = (π‘₯ ∈ 𝒫 β„•0, 𝑦 ∈ β„•0 ↦ (π‘₯ sadd {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)}))
63 ovex 7438 . . . 4 ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}) ∈ V
6449, 62, 63ovmpoa 7559 . . 3 (((π‘ƒβ€˜π‘) ∈ 𝒫 β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((π‘ƒβ€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))𝑁) = ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}))
6536, 1, 64syl2anc 584 . 2 (πœ‘ β†’ ((π‘ƒβ€˜π‘)(𝑝 ∈ 𝒫 β„•0, π‘š ∈ β„•0 ↦ (𝑝 sadd {𝑛 ∈ β„•0 ∣ (π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡)}))𝑁) = ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}))
6610, 32, 653eqtrd 2776 1 (πœ‘ β†’ (π‘ƒβ€˜(𝑁 + 1)) = ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  π’« cpw 4601   ↦ cmpt 5230  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  0cc0 11106  1c1 11107   + caddc 11109   βˆ’ cmin 11440  β„•cn 12208  β„•0cn0 12468  β„€β‰₯cuz 12818  seqcseq 13962   sadd csad 16357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-had 1595  df-cad 1608  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-seq 13963  df-sad 16388
This theorem is referenced by:  smuval2  16419  smupvallem  16420  smu01lem  16422  smupval  16425  smup1  16426  smueqlem  16427
  Copyright terms: Public domain W3C validator