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Mirrors > Home > MPE Home > Th. List > sadcf | Structured version Visualization version GIF version |
Description: The carry sequence is a sequence of elements of 2𝑜 encoding a "sequence of wffs". (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
sadval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ0) |
sadval.b | ⊢ (𝜑 → 𝐵 ⊆ ℕ0) |
sadval.c | ⊢ 𝐶 = seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) |
Ref | Expression |
---|---|
sadcf | ⊢ (𝜑 → 𝐶:ℕ0⟶2𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11184 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
2 | iftrue 4042 | . . . . . . 7 ⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) = ∅) | |
3 | eqid 2610 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) | |
4 | 0ex 4718 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 2, 3, 4 | fvmpt 6191 | . . . . . 6 ⊢ (0 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) = ∅) |
6 | 1, 5 | ax-mp 5 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) = ∅ |
7 | 4 | prid1 4241 | . . . . . 6 ⊢ ∅ ∈ {∅, 1𝑜} |
8 | df2o3 7460 | . . . . . 6 ⊢ 2𝑜 = {∅, 1𝑜} | |
9 | 7, 8 | eleqtrri 2687 | . . . . 5 ⊢ ∅ ∈ 2𝑜 |
10 | 6, 9 | eqeltri 2684 | . . . 4 ⊢ ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) ∈ 2𝑜 |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) ∈ 2𝑜) |
12 | df-ov 6552 | . . . . 5 ⊢ (𝑥(𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅))𝑦) = ((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅))‘〈𝑥, 𝑦〉) | |
13 | 1on 7454 | . . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On | |
14 | 13 | elexi 3186 | . . . . . . . . . . 11 ⊢ 1𝑜 ∈ V |
15 | 14 | prid2 4242 | . . . . . . . . . 10 ⊢ 1𝑜 ∈ {∅, 1𝑜} |
16 | 15, 8 | eleqtrri 2687 | . . . . . . . . 9 ⊢ 1𝑜 ∈ 2𝑜 |
17 | 16, 9 | keepel 4105 | . . . . . . . 8 ⊢ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅) ∈ 2𝑜 |
18 | 17 | rgen2w 2909 | . . . . . . 7 ⊢ ∀𝑐 ∈ 2𝑜 ∀𝑚 ∈ ℕ0 if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅) ∈ 2𝑜 |
19 | eqid 2610 | . . . . . . . 8 ⊢ (𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)) = (𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)) | |
20 | 19 | fmpt2 7126 | . . . . . . 7 ⊢ (∀𝑐 ∈ 2𝑜 ∀𝑚 ∈ ℕ0 if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅) ∈ 2𝑜 ↔ (𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)):(2𝑜 × ℕ0)⟶2𝑜) |
21 | 18, 20 | mpbi 219 | . . . . . 6 ⊢ (𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)):(2𝑜 × ℕ0)⟶2𝑜 |
22 | 21, 9 | f0cli 6278 | . . . . 5 ⊢ ((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅))‘〈𝑥, 𝑦〉) ∈ 2𝑜 |
23 | 12, 22 | eqeltri 2684 | . . . 4 ⊢ (𝑥(𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅))𝑦) ∈ 2𝑜 |
24 | 23 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 2𝑜 ∧ 𝑦 ∈ V)) → (𝑥(𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅))𝑦) ∈ 2𝑜) |
25 | nn0uz 11598 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
26 | 0zd 11266 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
27 | fvex 6113 | . . . 4 ⊢ ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V | |
28 | 27 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(0 + 1))) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V) |
29 | 11, 24, 25, 26, 28 | seqf2 12682 | . 2 ⊢ (𝜑 → seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))):ℕ0⟶2𝑜) |
30 | sadval.c | . . 3 ⊢ 𝐶 = seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) | |
31 | 30 | feq1i 5949 | . 2 ⊢ (𝐶:ℕ0⟶2𝑜 ↔ seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))):ℕ0⟶2𝑜) |
32 | 29, 31 | sylibr 223 | 1 ⊢ (𝜑 → 𝐶:ℕ0⟶2𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 caddwcad 1536 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {cpr 4127 〈cop 4131 ↦ cmpt 4643 × cxp 5036 Oncon0 5640 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1𝑜c1o 7440 2𝑜c2o 7441 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕ0cn0 11169 ℤ≥cuz 11563 seqcseq 12663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 |
This theorem is referenced by: sadcp1 15015 |
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