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Mirrors > Home > MPE Home > Th. List > sadval | Structured version Visualization version GIF version |
Description: The full adder sequence is the half adder function applied to the inputs and the carry sequence. (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)))) |
sadcp1.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
sadval | ⊢ (𝜑 → (𝑁 ∈ (𝐴 sadd 𝐵) ↔ hadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sadval.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℕ0) | |
2 | sadval.b | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℕ0) | |
3 | sadval.c | . . . 4 ⊢ 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) | |
4 | 1, 2, 3 | sadfval 15590 | . . 3 ⊢ (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))}) |
5 | 4 | eleq2d 2845 | . 2 ⊢ (𝜑 → (𝑁 ∈ (𝐴 sadd 𝐵) ↔ 𝑁 ∈ {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))})) |
6 | sadcp1.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
7 | eleq1 2847 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
8 | eleq1 2847 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ 𝐵 ↔ 𝑁 ∈ 𝐵)) | |
9 | fveq2 6448 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐶‘𝑘) = (𝐶‘𝑁)) | |
10 | 9 | eleq2d 2845 | . . . . 5 ⊢ (𝑘 = 𝑁 → (∅ ∈ (𝐶‘𝑘) ↔ ∅ ∈ (𝐶‘𝑁))) |
11 | 7, 8, 10 | hadbi123d 1653 | . . . 4 ⊢ (𝑘 = 𝑁 → (hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘)) ↔ hadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))) |
12 | 11 | elrab3 3574 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))} ↔ hadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))) |
13 | 6, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑁 ∈ {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))} ↔ hadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))) |
14 | 5, 13 | bitrd 271 | 1 ⊢ (𝜑 → (𝑁 ∈ (𝐴 sadd 𝐵) ↔ hadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 haddwhad 1651 caddwcad 1664 ∈ wcel 2107 {crab 3094 ⊆ 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 seqcseq 13124 sadd csad 15558 |
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-1cn 10332 ax-addcl 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-xor 1583 df-tru 1605 df-had 1652 df-cad 1665 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-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-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-nn 11380 df-n0 11648 df-seq 13125 df-sad 15589 |
This theorem is referenced by: sadadd2lem 15597 saddisjlem 15602 |
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