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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 11307 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
2 | iftrue 4092 | . . . . . . 7 ⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) = ∅) | |
3 | eqid 2622 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) | |
4 | 0ex 4790 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 2, 3, 4 | fvmpt 6282 | . . . . . 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 4297 | . . . . . 6 ⊢ ∅ ∈ {∅, 1𝑜} |
8 | df2o3 7573 | . . . . . 6 ⊢ 2𝑜 = {∅, 1𝑜} | |
9 | 7, 8 | eleqtrri 2700 | . . . . 5 ⊢ ∅ ∈ 2𝑜 |
10 | 6, 9 | eqeltri 2697 | . . . 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 6653 | . . . . 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 7567 | . . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On | |
14 | 13 | elexi 3213 | . . . . . . . . . . 11 ⊢ 1𝑜 ∈ V |
15 | 14 | prid2 4298 | . . . . . . . . . 10 ⊢ 1𝑜 ∈ {∅, 1𝑜} |
16 | 15, 8 | eleqtrri 2700 | . . . . . . . . 9 ⊢ 1𝑜 ∈ 2𝑜 |
17 | 16, 9 | keepel 4155 | . . . . . . . 8 ⊢ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅) ∈ 2𝑜 |
18 | 17 | rgen2w 2925 | . . . . . . 7 ⊢ ∀𝑐 ∈ 2𝑜 ∀𝑚 ∈ ℕ0 if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅) ∈ 2𝑜 |
19 | eqid 2622 | . . . . . . . 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 7237 | . . . . . . 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 6370 | . . . . 5 ⊢ ((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1𝑜, ∅))‘〈𝑥, 𝑦〉) ∈ 2𝑜 |
23 | 12, 22 | eqeltri 2697 | . . . 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 11722 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
26 | 0zd 11389 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
27 | fvexd 6203 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(0 + 1))) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V) | |
28 | 11, 24, 25, 26, 27 | seqf2 12820 | . 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 6036 | . 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 384 = wceq 1483 caddwcad 1545 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ifcif 4086 {cpr 4179 〈cop 4183 ↦ cmpt 4729 × cxp 5112 Oncon0 5723 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1𝑜c1o 7553 2𝑜c2o 7554 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 ℕ0cn0 11292 ℤ≥cuz 11687 seqcseq 12801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 |
This theorem is referenced by: sadcp1 15177 |
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