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Theorem smupp1 16426
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 12865 . . . . 5 β„•0 = (β„€β‰₯β€˜0)
31, 2eleqtrdi 2837 . . . 4 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
4 seqp1 13984 . . . 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 6885 . . 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 6885 . . . 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 7414 . . 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 2791 . 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 12489 . . . . . . 7 1 ∈ β„•0
1211a1i 11 . . . . . 6 (πœ‘ β†’ 1 ∈ β„•0)
131, 12nn0addcld 12537 . . . . 5 (πœ‘ β†’ (𝑁 + 1) ∈ β„•0)
14 eqeq1 2730 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 7411 . . . . . . 7 (𝑛 = (𝑁 + 1) β†’ (𝑛 βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
1614, 15ifbieq2d 4549 . . . . . 6 (𝑛 = (𝑁 + 1) β†’ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)) = if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)))
17 eqid 2726 . . . . . 6 (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1))) = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))
18 0ex 5300 . . . . . . 7 βˆ… ∈ V
19 ovex 7437 . . . . . . 7 ((𝑁 + 1) βˆ’ 1) ∈ V
2018, 19ifex 4573 . . . . . 6 if((𝑁 + 1) = 0, βˆ…, ((𝑁 + 1) βˆ’ 1)) ∈ V
2116, 17, 20fvmpt 6991 . . . . 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 12512 . . . . . . 7 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•)
241, 23syl 17 . . . . . 6 (πœ‘ β†’ (𝑁 + 1) ∈ β„•)
2524nnne0d 12263 . . . . 5 (πœ‘ β†’ (𝑁 + 1) β‰  0)
26 ifnefalse 4535 . . . . 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 12535 . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„‚)
2912nn0cnd 12535 . . . . 5 (πœ‘ β†’ 1 ∈ β„‚)
3028, 29pncand 11573 . . . 4 (πœ‘ β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
3122, 27, 303eqtrd 2770 . . 3 (πœ‘ β†’ ((𝑛 ∈ β„•0 ↦ if(𝑛 = 0, βˆ…, (𝑛 βˆ’ 1)))β€˜(𝑁 + 1)) = 𝑁)
3231oveq2d 7420 . 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 16424 . . . 4 (πœ‘ β†’ 𝑃:β„•0βŸΆπ’« β„•0)
3635, 1ffvelcdmd 7080 . . 3 (πœ‘ β†’ (π‘ƒβ€˜π‘) ∈ 𝒫 β„•0)
37 simpl 482 . . . . 5 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ π‘₯ = (π‘ƒβ€˜π‘))
38 simpr 484 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ 𝑦 = 𝑁)
3938eleq1d 2812 . . . . . . . 8 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
4038oveq2d 7420 . . . . . . . . 9 ((π‘₯ = (π‘ƒβ€˜π‘) ∧ 𝑦 = 𝑁) β†’ (π‘˜ βˆ’ 𝑦) = (π‘˜ βˆ’ 𝑁))
4140eleq1d 2812 . . . . . . . 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 3434 . . . . . 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 7411 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑁) = (𝑛 βˆ’ 𝑁))
4544eleq1d 2812 . . . . . . . 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 3436 . . . . . 6 {π‘˜ ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑁) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}
4843, 47eqtrdi 2782 . . . . 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 7422 . . . 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 7411 . . . . 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 2810 . . . . . . . . 9 (π‘š = 𝑦 β†’ (π‘š ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
52 oveq2 7412 . . . . . . . . . 10 (π‘š = 𝑦 β†’ (𝑛 βˆ’ π‘š) = (𝑛 βˆ’ 𝑦))
5352eleq1d 2812 . . . . . . . . 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 3434 . . . . . . 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 7411 . . . . . . . . . 10 (π‘˜ = 𝑛 β†’ (π‘˜ βˆ’ 𝑦) = (𝑛 βˆ’ 𝑦))
5756eleq1d 2812 . . . . . . . . 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 3436 . . . . . . 7 {π‘˜ ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (π‘˜ βˆ’ 𝑦) ∈ 𝐡)} = {𝑛 ∈ β„•0 ∣ (𝑦 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑦) ∈ 𝐡)}
6055, 59eqtr4di 2784 . . . . . 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 7420 . . . . 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 7499 . . . 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 7437 . . . 4 ((π‘ƒβ€˜π‘) sadd {𝑛 ∈ β„•0 ∣ (𝑁 ∈ 𝐴 ∧ (𝑛 βˆ’ 𝑁) ∈ 𝐡)}) ∈ V
6449, 62, 63ovmpoa 7558 . . 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 583 . 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 2770 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426   βŠ† wss 3943  βˆ…c0 4317  ifcif 4523  π’« cpw 4597   ↦ cmpt 5224  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  0cc0 11109  1c1 11110   + caddc 11112   βˆ’ cmin 11445  β„•cn 12213  β„•0cn0 12473  β„€β‰₯cuz 12823  seqcseq 13969   sadd csad 16366
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-seq 13970  df-sad 16397
This theorem is referenced by:  smuval2  16428  smupvallem  16429  smu01lem  16431  smupval  16434  smup1  16435  smueqlem  16436
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