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Theorem smupp1 16367
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 12812 . . . . 5 β„•0 = (β„€β‰₯β€˜0)
31, 2eleqtrdi 2848 . . . 4 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
4 seqp1 13928 . . . 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 6848 . . 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 6848 . . . 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 7372 . . 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 2802 . 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 12436 . . . . . . 7 1 ∈ β„•0
1211a1i 11 . . . . . 6 (πœ‘ β†’ 1 ∈ β„•0)
131, 12nn0addcld 12484 . . . . 5 (πœ‘ β†’ (𝑁 + 1) ∈ β„•0)
14 eqeq1 2741 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 7369 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
1614, 15ifbieq2d 4517 . . . . . 6 (𝑛 = (𝑁 + 1) β†’ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)) = if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)))
17 eqid 2737 . . . . . 6 (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))) = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))
18 0ex 5269 . . . . . . 7 βˆ… ∈ V
19 ovex 7395 . . . . . . 7 ((𝑁 + 1) βˆ’ 1) ∈ V
2018, 19ifex 4541 . . . . . 6 if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)) ∈ V
2116, 17, 20fvmpt 6953 . . . . 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 12459 . . . . . . 7 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
241, 23syl 17 . . . . . 6 (πœ‘ β†’ (𝑁 + 1) ∈ β„•)
2524nnne0d 12210 . . . . 5 (πœ‘ β†’ (𝑁 + 1) β‰  0)
26 ifnefalse 4503 . . . . 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 12482 . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„‚)
2912nn0cnd 12482 . . . . 5 (πœ‘ β†’ 1 ∈ β„‚)
3028, 29pncand 11520 . . . 4 (πœ‘ β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
3122, 27, 303eqtrd 2781 . . 3 (πœ‘ β†’ ((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1)) = 𝑁)
3231oveq2d 7378 . 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 16365 . . . 4 (πœ‘ β†’ 𝑃:β„•0βŸΆπ’« β„•0)
3635, 1ffvelcdmd 7041 . . 3 (πœ‘ β†’ (π‘ƒβ€˜π‘) ∈ 𝒫 β„•0)
37 simpl 484 . . . . 5 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ π‘₯ = (π‘ƒβ€˜π‘))
38 simpr 486 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ 𝑦 = 𝑁)
3938eleq1d 2823 . . . . . . . 8 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
4038oveq2d 7378 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (π‘˜ βˆ’ 𝑦) = (π‘˜ βˆ’ 𝑁))
4140eleq1d 2823 . . . . . . . 8 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ ((π‘˜ βˆ’ 𝑦) ∈ 𝐡 ↔ (π‘˜ βˆ’ 𝑁) ∈ 𝐡))
4239, 41anbi12d 632 . . . . . . 7 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ ((𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡) ↔ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)))
4342rabbidv 3418 . . . . . 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 7369 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑁) = (𝑛 βˆ’ 𝑁))
4544eleq1d 2823 . . . . . . . 8 (π‘˜ = 𝑛 β†’ ((π‘˜ βˆ’ 𝑁) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑁) ∈ 𝐡))
4645anbi2d 630 . . . . . . 7 (π‘˜ = 𝑛 β†’ ((𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡) ↔ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)))
4746cbvrabv 3420 . . . . . 6 {π‘˜ ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}
4843, 47eqtrdi 2793 . . . . 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 7380 . . . 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 7369 . . . . 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 2821 . . . . . . . . 9 (π‘š = 𝑦 β†’ (π‘š ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
52 oveq2 7370 . . . . . . . . . 10 (π‘š = 𝑦 β†’ (𝑛 βˆ’ π‘š) = (𝑛 βˆ’ 𝑦))
5352eleq1d 2823 . . . . . . . . 9 (π‘š = 𝑦 β†’ ((𝑛 βˆ’ π‘š) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑦) ∈ 𝐡))
5451, 53anbi12d 632 . . . . . . . 8 (π‘š = 𝑦 β†’ ((π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)))
5554rabbidv 3418 . . . . . . 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 7369 . . . . . . . . . 10 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑦) = (𝑛 βˆ’ 𝑦))
5756eleq1d 2823 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ ((π‘˜ βˆ’ 𝑦) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑦) ∈ 𝐡))
5857anbi2d 630 . . . . . . . 8 (π‘˜ = 𝑛 β†’ ((𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)))
5958cbvrabv 3420 . . . . . . 7 {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)}
6055, 59eqtr4di 2795 . . . . . 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 7378 . . . . 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 7457 . . . 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 7395 . . . 4 ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}) ∈ V
6449, 62, 63ovmpoa 7515 . . 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 585 . 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 2781 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  {crab 3410   βŠ† wss 3915  βˆ…c0 4287  ifcif 4491  π’« cpw 4565   ↦ cmpt 5193  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  0cc0 11058  1c1 11059   + caddc 11061   βˆ’ cmin 11392  β„•cn 12160  β„•0cn0 12420  β„€β‰₯cuz 12770  seqcseq 13913   sadd csad 16307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-xor 1511  df-tru 1545  df-fal 1555  df-had 1596  df-cad 1609  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  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 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-seq 13914  df-sad 16338
This theorem is referenced by:  smuval2  16369  smupvallem  16370  smu01lem  16372  smupval  16375  smup1  16376  smueqlem  16377
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