![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 16398 | . . 3 ⊢ (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))}) |
5 | 4 | eleq2d 2813 | . 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 2815 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
8 | eleq1 2815 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ 𝐵 ↔ 𝑁 ∈ 𝐵)) | |
9 | fveq2 6884 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐶‘𝑘) = (𝐶‘𝑁)) | |
10 | 9 | eleq2d 2813 | . . . . 5 ⊢ (𝑘 = 𝑁 → (∅ ∈ (𝐶‘𝑘) ↔ ∅ ∈ (𝐶‘𝑁))) |
11 | 7, 8, 10 | hadbi123d 1588 | . . . 4 ⊢ (𝑘 = 𝑁 → (hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘)) ↔ hadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))) |
12 | 11 | elrab3 3679 | . . 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 279 | 1 ⊢ (𝜑 → (𝑁 ∈ (𝐴 sadd 𝐵) ↔ hadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 haddwhad 1586 caddwcad 1599 ∈ wcel 2098 {crab 3426 ⊆ wss 3943 ∅c0 4317 ifcif 4523 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 1oc1o 8457 2oc2o 8458 0cc0 11109 1c1 11110 − cmin 11445 ℕ0cn0 12473 seqcseq 13969 sadd csad 16366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-had 1587 df-cad 1600 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-nn 12214 df-n0 12474 df-seq 13970 df-sad 16397 |
This theorem is referenced by: sadadd2lem 16405 saddisjlem 16410 |
Copyright terms: Public domain | W3C validator |