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Theorem sadfval 16258
Description: Define the addition of two bit sequences, using df-had 1594 and df-cad 1607 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)
Hypotheses
Ref Expression
sadval.a (𝜑𝐴 ⊆ ℕ0)
sadval.b (𝜑𝐵 ⊆ ℕ0)
sadval.c 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
Assertion
Ref Expression
sadfval (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))})
Distinct variable groups:   𝑘,𝑐,𝑚,𝑛   𝐴,𝑐,𝑘,𝑚   𝐵,𝑐,𝑘,𝑚   𝐶,𝑘   𝜑,𝑘
Allowed substitution hints:   𝜑(𝑚,𝑛,𝑐)   𝐴(𝑛)   𝐵(𝑛)   𝐶(𝑚,𝑛,𝑐)

Proof of Theorem sadfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sadval.a . . 3 (𝜑𝐴 ⊆ ℕ0)
2 nn0ex 12340 . . . 4 0 ∈ V
32elpw2 5289 . . 3 (𝐴 ∈ 𝒫 ℕ0𝐴 ⊆ ℕ0)
41, 3sylibr 233 . 2 (𝜑𝐴 ∈ 𝒫 ℕ0)
5 sadval.b . . 3 (𝜑𝐵 ⊆ ℕ0)
62elpw2 5289 . . 3 (𝐵 ∈ 𝒫 ℕ0𝐵 ⊆ ℕ0)
75, 6sylibr 233 . 2 (𝜑𝐵 ∈ 𝒫 ℕ0)
8 simpl 483 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
98eleq2d 2822 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘𝑥𝑘𝐴))
10 simpr 485 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
1110eleq2d 2822 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘𝑦𝑘𝐵))
12 simp1l 1196 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
1312eleq2d 2822 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (𝑚𝑥𝑚𝐴))
14 simp1r 1197 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
1514eleq2d 2822 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (𝑚𝑦𝑚𝐵))
16 biidd 261 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (∅ ∈ 𝑐 ↔ ∅ ∈ 𝑐))
1713, 15, 16cadbi123d 1610 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐) ↔ cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐)))
1817ifbid 4496 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅))
1918mpoeq3dva 7414 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)) = (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅)))
2019seqeq2d 13829 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))))
21 sadval.c . . . . . . . 8 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
2220, 21eqtr4di 2794 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = 𝐶)
2322fveq1d 6827 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = (𝐶𝑘))
2423eleq2d 2822 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) ↔ ∅ ∈ (𝐶𝑘)))
259, 11, 24hadbi123d 1595 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (hadd(𝑘𝑥, 𝑘𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)) ↔ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))))
2625rabbidv 3411 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝑥, 𝑘𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))} = {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))})
27 df-sad 16257 . . 3 sadd = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝑥, 𝑘𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
282rabex 5276 . . 3 {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))} ∈ V
2926, 27, 28ovmpoa 7490 . 2 ((𝐴 ∈ 𝒫 ℕ0𝐵 ∈ 𝒫 ℕ0) → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))})
304, 7, 29syl2anc 584 1 (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  haddwhad 1593  caddwcad 1606  wcel 2105  {crab 3403  wss 3898  c0 4269  ifcif 4473  𝒫 cpw 4547  cmpt 5175  cfv 6479  (class class class)co 7337  cmpo 7339  1oc1o 8360  2oc2o 8361  0cc0 10972  1c1 10973  cmin 11306  0cn0 12334  seqcseq 13822   sadd csad 16226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-1cn 11030  ax-addcl 11032
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1543  df-fal 1553  df-had 1594  df-cad 1607  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-nn 12075  df-n0 12335  df-seq 13823  df-sad 16257
This theorem is referenced by:  sadval  16262  sadadd2lem  16265  sadadd3  16267  sadcl  16268  sadcom  16269
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