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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 11470 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
2 | iftrue 4224 | . . . . . . 7 ⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) = ∅) | |
3 | eqid 2748 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) | |
4 | 0ex 4930 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 2, 3, 4 | fvmpt 6432 | . . . . . 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 4429 | . . . . . 6 ⊢ ∅ ∈ {∅, 1𝑜} |
8 | df2o3 7730 | . . . . . 6 ⊢ 2𝑜 = {∅, 1𝑜} | |
9 | 7, 8 | eleqtrri 2826 | . . . . 5 ⊢ ∅ ∈ 2𝑜 |
10 | 6, 9 | eqeltri 2823 | . . . 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 6804 | . . . . 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 7724 | . . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On | |
14 | 13 | elexi 3341 | . . . . . . . . . . 11 ⊢ 1𝑜 ∈ V |
15 | 14 | prid2 4430 | . . . . . . . . . 10 ⊢ 1𝑜 ∈ {∅, 1𝑜} |
16 | 15, 8 | eleqtrri 2826 | . . . . . . . . 9 ⊢ 1𝑜 ∈ 2𝑜 |
17 | 16, 9 | keepel 4287 | . . . . . . . 8 ⊢ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅) ∈ 2𝑜 |
18 | 17 | rgen2w 3051 | . . . . . . 7 ⊢ ∀𝑐 ∈ 2𝑜 ∀𝑚 ∈ ℕ0 if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅) ∈ 2𝑜 |
19 | eqid 2748 | . . . . . . . 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 7393 | . . . . . . 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 220 | . . . . . 6 ⊢ (𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)):(2𝑜 × ℕ0)⟶2𝑜 |
22 | 21, 9 | f0cli 6521 | . . . . 5 ⊢ ((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅))‘〈𝑥, 𝑦〉) ∈ 2𝑜 |
23 | 12, 22 | eqeltri 2823 | . . . 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 11886 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
26 | 0zd 11552 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
27 | fvexd 6352 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(0 + 1))) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V) | |
28 | 11, 24, 25, 26, 27 | seqf2 12985 | . 2 ⊢ (𝜑 → seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))):ℕ0⟶2𝑜) |
29 | sadval.c | . . 3 ⊢ 𝐶 = seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) | |
30 | 29 | feq1i 6185 | . 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) |
31 | 28, 30 | sylibr 224 | 1 ⊢ (𝜑 → 𝐶:ℕ0⟶2𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 caddwcad 1682 ∈ wcel 2127 ∀wral 3038 Vcvv 3328 ⊆ wss 3703 ∅c0 4046 ifcif 4218 {cpr 4311 〈cop 4315 ↦ cmpt 4869 × cxp 5252 Oncon0 5872 ⟶wf 6033 ‘cfv 6037 (class class class)co 6801 ↦ cmpt2 6803 1𝑜c1o 7710 2𝑜c2o 7711 0cc0 10099 1c1 10100 + caddc 10102 − cmin 10429 ℕ0cn0 11455 ℤ≥cuz 11850 seqcseq 12966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-n0 11456 df-z 11541 df-uz 11851 df-fz 12491 df-seq 12967 |
This theorem is referenced by: sadcp1 15350 |
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