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Theorem smupp1 16462
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 12902 . . . . 5 β„•0 = (β„€β‰₯β€˜0)
31, 2eleqtrdi 2839 . . . 4 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
4 seqp1 14021 . . . 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 6903 . . 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 6903 . . . 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 7436 . . 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 2793 . 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 12526 . . . . . . 7 1 ∈ β„•0
1211a1i 11 . . . . . 6 (πœ‘ β†’ 1 ∈ β„•0)
131, 12nn0addcld 12574 . . . . 5 (πœ‘ β†’ (𝑁 + 1) ∈ β„•0)
14 eqeq1 2732 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 7433 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
1614, 15ifbieq2d 4558 . . . . . 6 (𝑛 = (𝑁 + 1) β†’ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)) = if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)))
17 eqid 2728 . . . . . 6 (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))) = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))
18 0ex 5311 . . . . . . 7 βˆ… ∈ V
19 ovex 7459 . . . . . . 7 ((𝑁 + 1) βˆ’ 1) ∈ V
2018, 19ifex 4582 . . . . . 6 if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)) ∈ V
2116, 17, 20fvmpt 7010 . . . . 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 12549 . . . . . . 7 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
241, 23syl 17 . . . . . 6 (πœ‘ β†’ (𝑁 + 1) ∈ β„•)
2524nnne0d 12300 . . . . 5 (πœ‘ β†’ (𝑁 + 1) β‰  0)
26 ifnefalse 4544 . . . . 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 12572 . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„‚)
2912nn0cnd 12572 . . . . 5 (πœ‘ β†’ 1 ∈ β„‚)
3028, 29pncand 11610 . . . 4 (πœ‘ β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
3122, 27, 303eqtrd 2772 . . 3 (πœ‘ β†’ ((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1)) = 𝑁)
3231oveq2d 7442 . 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 16460 . . . 4 (πœ‘ β†’ 𝑃:β„•0βŸΆπ’« β„•0)
3635, 1ffvelcdmd 7100 . . 3 (πœ‘ β†’ (π‘ƒβ€˜π‘) ∈ 𝒫 β„•0)
37 simpl 481 . . . . 5 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ π‘₯ = (π‘ƒβ€˜π‘))
38 simpr 483 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ 𝑦 = 𝑁)
3938eleq1d 2814 . . . . . . . 8 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
4038oveq2d 7442 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (π‘˜ βˆ’ 𝑦) = (π‘˜ βˆ’ 𝑁))
4140eleq1d 2814 . . . . . . . 8 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ ((π‘˜ βˆ’ 𝑦) ∈ 𝐡 ↔ (π‘˜ βˆ’ 𝑁) ∈ 𝐡))
4239, 41anbi12d 630 . . . . . . 7 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ ((𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡) ↔ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)))
4342rabbidv 3438 . . . . . 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 7433 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑁) = (𝑛 βˆ’ 𝑁))
4544eleq1d 2814 . . . . . . . 8 (π‘˜ = 𝑛 β†’ ((π‘˜ βˆ’ 𝑁) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑁) ∈ 𝐡))
4645anbi2d 628 . . . . . . 7 (π‘˜ = 𝑛 β†’ ((𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡) ↔ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)))
4746cbvrabv 3441 . . . . . 6 {π‘˜ ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}
4843, 47eqtrdi 2784 . . . . 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 7444 . . . 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 7433 . . . . 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 2812 . . . . . . . . 9 (π‘š = 𝑦 β†’ (π‘š ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
52 oveq2 7434 . . . . . . . . . 10 (π‘š = 𝑦 β†’ (𝑛 βˆ’ π‘š) = (𝑛 βˆ’ 𝑦))
5352eleq1d 2814 . . . . . . . . 9 (π‘š = 𝑦 β†’ ((𝑛 βˆ’ π‘š) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑦) ∈ 𝐡))
5451, 53anbi12d 630 . . . . . . . 8 (π‘š = 𝑦 β†’ ((π‘š ∈ 𝐴 ∧ (𝑛 βˆ’ π‘š) ∈ 𝐡) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)))
5554rabbidv 3438 . . . . . . 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 7433 . . . . . . . . . 10 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑦) = (𝑛 βˆ’ 𝑦))
5756eleq1d 2814 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ ((π‘˜ βˆ’ 𝑦) ∈ 𝐡 ↔ (𝑛 βˆ’ 𝑦) ∈ 𝐡))
5857anbi2d 628 . . . . . . . 8 (π‘˜ = 𝑛 β†’ ((𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡) ↔ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)))
5958cbvrabv 3441 . . . . . . 7 {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)}
6055, 59eqtr4di 2786 . . . . . 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 7442 . . . . 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 7521 . . . 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 7459 . . . 4 ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}) ∈ V
6449, 62, 63ovmpoa 7582 . . 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 582 . 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 2772 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 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  {crab 3430   βŠ† wss 3949  βˆ…c0 4326  ifcif 4532  π’« cpw 4606   ↦ cmpt 5235  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  0cc0 11146  1c1 11147   + caddc 11149   βˆ’ cmin 11482  β„•cn 12250  β„•0cn0 12510  β„€β‰₯cuz 12860  seqcseq 14006   sadd csad 16402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-had 1587  df-cad 1600  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-seq 14007  df-sad 16433
This theorem is referenced by:  smuval2  16464  smupvallem  16465  smu01lem  16467  smupval  16470  smup1  16471  smueqlem  16472
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