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Theorem sadfval 16434
Description: Define the addition of two bit sequences, using df-had 1587 and df-cad 1600 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 12516 . . . 4 0 ∈ V
32elpw2 5351 . . 3 (𝐴 ∈ 𝒫 ℕ0𝐴 ⊆ ℕ0)
41, 3sylibr 233 . 2 (𝜑𝐴 ∈ 𝒫 ℕ0)
5 sadval.b . . 3 (𝜑𝐵 ⊆ ℕ0)
62elpw2 5351 . . 3 (𝐵 ∈ 𝒫 ℕ0𝐵 ⊆ ℕ0)
75, 6sylibr 233 . 2 (𝜑𝐵 ∈ 𝒫 ℕ0)
8 simpl 481 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
98eleq2d 2815 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘𝑥𝑘𝐴))
10 simpr 483 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
1110eleq2d 2815 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘𝑦𝑘𝐵))
12 simp1l 1194 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
1312eleq2d 2815 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (𝑚𝑥𝑚𝐴))
14 simp1r 1195 . . . . . . . . . . . . 13 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
1514eleq2d 2815 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (𝑚𝑦𝑚𝐵))
16 biidd 261 . . . . . . . . . . . 12 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (∅ ∈ 𝑐 ↔ ∅ ∈ 𝑐))
1713, 15, 16cadbi123d 1603 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → (cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐) ↔ cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐)))
1817ifbid 4555 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑐 ∈ 2o𝑚 ∈ ℕ0) → if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚𝐴, 𝑚𝐵, ∅ ∈ 𝑐), 1o, ∅))
1918mpoeq3dva 7503 . . . . . . . . 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 14013 . . . . . . . 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 2786 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = 𝐶)
2322fveq1d 6904 . . . . . 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 2815 . . . . 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 1588 . . . 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 3438 . . 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 16433 . . 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 5338 . . 3 {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))} ∈ V
2926, 27, 28ovmpoa 7582 . 2 ((𝐴 ∈ 𝒫 ℕ0𝐵 ∈ 𝒫 ℕ0) → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))})
304, 7, 29syl2anc 582 1 (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝐴, 𝑘𝐵, ∅ ∈ (𝐶𝑘))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  haddwhad 1586  caddwcad 1599  wcel 2098  {crab 3430  wss 3949  c0 4326  ifcif 4532  𝒫 cpw 4606  cmpt 5235  cfv 6553  (class class class)co 7426  cmpo 7428  1oc1o 8486  2oc2o 8487  0cc0 11146  1c1 11147  cmin 11482  0cn0 12510  seqcseq 14006   sadd csad 16402
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-1cn 11204  ax-addcl 11206
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-nn 12251  df-n0 12511  df-seq 14007  df-sad 16433
This theorem is referenced by:  sadval  16438  sadadd2lem  16441  sadadd3  16443  sadcl  16444  sadcom  16445
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