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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 12425 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
2 | iftrue 4491 | . . . . . . 7 ⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) = ∅) | |
3 | eqid 2736 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) | |
4 | 0ex 5263 | . . . . . . 7 ⊢ ∅ ∈ V | |
5 | 2, 3, 4 | fvmpt 6946 | . . . . . 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 4722 | . . . . . 6 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8417 | . . . . . 6 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2837 | . . . . 5 ⊢ ∅ ∈ 2o |
10 | 6, 9 | eqeltri 2834 | . . . 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 7357 | . . . . 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 | 1oex 8419 | . . . . . . . . . . 11 ⊢ 1o ∈ V | |
14 | 13 | prid2 4723 | . . . . . . . . . 10 ⊢ 1o ∈ {∅, 1o} |
15 | 14, 8 | eleqtrri 2837 | . . . . . . . . 9 ⊢ 1o ∈ 2o |
16 | 15, 9 | ifcli 4532 | . . . . . . . 8 ⊢ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅) ∈ 2o |
17 | 16 | rgen2w 3068 | . . . . . . 7 ⊢ ∀𝑐 ∈ 2o ∀𝑚 ∈ ℕ0 if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅) ∈ 2o |
18 | eqid 2736 | . . . . . . . 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, ∅)) | |
19 | 18 | fmpo 7997 | . . . . . . 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) |
20 | 17, 19 | mpbi 229 | . . . . . 6 ⊢ (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)):(2o × ℕ0)⟶2o |
21 | 20, 9 | f0cli 7045 | . . . . 5 ⊢ ((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))‘〈𝑥, 𝑦〉) ∈ 2o |
22 | 12, 21 | eqeltri 2834 | . . . 4 ⊢ (𝑥(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))𝑦) ∈ 2o |
23 | 22 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 2o ∧ 𝑦 ∈ V)) → (𝑥(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))𝑦) ∈ 2o) |
24 | nn0uz 12802 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
25 | 0zd 12508 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
26 | fvexd 6855 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(0 + 1))) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V) | |
27 | 11, 23, 24, 25, 26 | seqf2 13924 | . 2 ⊢ (𝜑 → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))):ℕ0⟶2o) |
28 | sadval.c | . . 3 ⊢ 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) | |
29 | 28 | feq1i 6657 | . 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) |
30 | 27, 29 | sylibr 233 | 1 ⊢ (𝜑 → 𝐶:ℕ0⟶2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 caddwcad 1607 ∈ wcel 2106 ∀wral 3063 Vcvv 3444 ⊆ wss 3909 ∅c0 4281 ifcif 4485 {cpr 4587 〈cop 4591 ↦ cmpt 5187 × cxp 5630 ⟶wf 6490 ‘cfv 6494 (class class class)co 7354 ∈ cmpo 7356 1oc1o 8402 2oc2o 8403 0cc0 11048 1c1 11049 + caddc 11051 − cmin 11382 ℕ0cn0 12410 ℤ≥cuz 12760 seqcseq 13903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-n0 12411 df-z 12497 df-uz 12761 df-fz 13422 df-seq 13904 |
This theorem is referenced by: sadcp1 16332 |
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