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Theorem sadfval 16486
Description: Define the addition of two bit sequences, using df-had 1591 and df-cad 1604 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 12530 . . . 4 0 ∈ V
32elpw2 5340 . . 3 (𝐴 ∈ 𝒫 ℕ0𝐴 ⊆ ℕ0)
41, 3sylibr 234 . 2 (𝜑𝐴 ∈ 𝒫 ℕ0)
5 sadval.b . . 3 (𝜑𝐵 ⊆ ℕ0)
62elpw2 5340 . . 3 (𝐵 ∈ 𝒫 ℕ0𝐵 ⊆ ℕ0)
75, 6sylibr 234 . 2 (𝜑𝐵 ∈ 𝒫 ℕ0)
8 simpl 482 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
98eleq2d 2825 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘𝑥𝑘𝐴))
10 simpr 484 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
1110eleq2d 2825 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘𝑦𝑘𝐵))
12 simp1l 1196 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
1312eleq2d 2825 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (𝑚𝑥𝑚𝐴))
14 simp1r 1197 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
1514eleq2d 2825 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (𝑚𝑦𝑚𝐵))
16 biidd 262 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (∅ ∈ 𝑐 ↔ ∅ ∈ 𝑐))
1713, 15, 16cadbi123d 1607 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐) ↔ cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐)))
1817ifbid 4554 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅))
1918mpoeq3dva 7510 . . . . . . . . 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 14046 . . . . . . . 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 2793 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = 𝐶)
2322fveq1d 6909 . . . . . 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 2825 . . . . 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 1592 . . . 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 3441 . . 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 16485 . . 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 5345 . . 3 {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))} ∈ V
2926, 27, 28ovmpoa 7588 . 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 395  w3a 1086   = wceq 1537  haddwhad 1590  caddwcad 1603  wcel 2106  {crab 3433  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  cmpt 5231  cfv 6563  (class class class)co 7431  cmpo 7433  1oc1o 8498  2oc2o 8499  0cc0 11153  1c1 11154  cmin 11490  0cn0 12524  seqcseq 14039   sadd csad 16454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1540  df-fal 1550  df-had 1591  df-cad 1604  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-n0 12525  df-seq 14040  df-sad 16485
This theorem is referenced by:  sadval  16490  sadadd2lem  16493  sadadd3  16495  sadcl  16496  sadcom  16497
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