Theorem sadfval 16469

Description: Define the addition of two bit sequences, using df-had 1613 and df-cad 1626 bit operations. (Contributed by Mario Carneiro, 5-Sep-2016.)

Hypotheses

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))))

Assertion

Ref	Expression
sadfval	⊢ (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))})

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.

Step	Hyp	Ref	Expression
1		sadval.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ0)
2		nn0ex 12484	. . . 4 ⊢ ℕ0 ∈ V
3	2	elpw2 5289	. . 3 ⊢ (𝐴 ∈ 𝒫 ℕ0 ↔ 𝐴 ⊆ ℕ0)
4	1, 3	sylibr 236	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ℕ0)
5		sadval.b	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℕ0)
6	2	elpw2 5289	. . 3 ⊢ (𝐵 ∈ 𝒫 ℕ0 ↔ 𝐵 ⊆ ℕ0)
7	5, 6	sylibr 236	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝒫 ℕ0)
8		simpl 486	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
9	8	eleq2d 2847	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑘 ∈ 𝑥 ↔ 𝑘 ∈ 𝐴))
10		simpr 488	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
11	10	eleq2d 2847	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑘 ∈ 𝑦 ↔ 𝑘 ∈ 𝐵))
12		simp1l 1210	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → 𝑥 = 𝐴)
13	12	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ 𝑥 ↔ 𝑚 ∈ 𝐴))
14		simp1r 1211	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → 𝑦 = 𝐵)
15	14	eleq2d 2847	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ 𝑦 ↔ 𝑚 ∈ 𝐵))
16		biidd 264	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → (∅ ∈ 𝑐 ↔ ∅ ∈ 𝑐))
17	13, 15, 16	cadbi123d 1629	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → (cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐) ↔ cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐)))
18	17	ifbid 4503	. . . . . . . . . 10 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))
19	18	mpoeq3dva 7469	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)) = (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)))
20	19	seqeq2d 14018	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))))
21		sadval.c	. . . . . . . 8 ⊢ 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
22	20, 21	eqtr4di 2814	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = 𝐶)
23	22	fveq1d 6865	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = (𝐶‘𝑘))
24	23	eleq2d 2847	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) ↔ ∅ ∈ (𝐶‘𝑘)))
25	9, 11, 24	hadbi123d 1614	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)) ↔ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))))
26	25	rabbidv 3420	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))} = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))})
27		df-sad 16468	. . 3 ⊢ sadd = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
28	2	rabex 5294	. . 3 ⊢ {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))} ∈ V
29	26, 27, 28	ovmpoa 7547	. 2 ⊢ ((𝐴 ∈ 𝒫 ℕ0 ∧ 𝐵 ∈ 𝒫 ℕ0) → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))})
30	4, 7, 29	syl2anc 593	1 ⊢ (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))})

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559  haddwhad 1612  caddwcad 1625   ∈ wcel 2141  {crab 3413   ⊆ wss 3904  ∅c0 4285  ifcif 4479  𝒫 cpw 4554   ↦ cmpt 5180  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  1_oc1o 8425  2_oc2o 8426  0cc0 11070  1c1 11071   − cmin 11411  ℕ₀cn0 12478  seqcseq 14011   sadd csad 16437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-1cn 11128  ax-addcl 11130

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-xor 1531  df-tru 1562  df-fal 1572  df-had 1613  df-cad 1626  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-nn 12208  df-n0 12479  df-seq 14012  df-sad 16468

This theorem is referenced by:  sadval  16473  sadadd2lem  16476  sadadd3  16478  sadcl  16479  sadcom  16480