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Mirrors > Home > MPE Home > Th. List > sadc0 | Structured version Visualization version GIF version |
Description: The initial element of the carry sequence is ⊥. (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 |
---|---|
sadc0 | ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4146 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | sadval.c | . . . . . 6 ⊢ 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) | |
3 | 2 | fveq1i 6449 | . . . . 5 ⊢ (𝐶‘0) = (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) |
4 | 0z 11744 | . . . . . 6 ⊢ 0 ∈ ℤ | |
5 | seq1 13137 | . . . . . 6 ⊢ (0 ∈ ℤ → (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) = ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0)) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) = ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) |
7 | 0nn0 11664 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
8 | iftrue 4313 | . . . . . . 7 ⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) = ∅) | |
9 | eqid 2778 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) | |
10 | 0ex 5028 | . . . . . . 7 ⊢ ∅ ∈ V | |
11 | 8, 9, 10 | fvmpt 6544 | . . . . . 6 ⊢ (0 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) = ∅) |
12 | 7, 11 | ax-mp 5 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) = ∅ |
13 | 3, 6, 12 | 3eqtri 2806 | . . . 4 ⊢ (𝐶‘0) = ∅ |
14 | 13 | eleq2i 2851 | . . 3 ⊢ (∅ ∈ (𝐶‘0) ↔ ∅ ∈ ∅) |
15 | 1, 14 | mtbir 315 | . 2 ⊢ ¬ ∅ ∈ (𝐶‘0) |
16 | 15 | a1i 11 | 1 ⊢ (𝜑 → ¬ ∅ ∈ (𝐶‘0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1601 caddwcad 1664 ∈ wcel 2107 ⊆ wss 3792 ∅c0 4141 ifcif 4307 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 1oc1o 7838 2oc2o 7839 0cc0 10274 1c1 10275 − cmin 10608 ℕ0cn0 11647 ℤcz 11733 seqcseq 13124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 df-seq 13125 |
This theorem is referenced by: sadcadd 15596 sadadd2 15598 saddisjlem 15602 |
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