Theorem sadcom 16388

Description: The adder sequence function is commutative. (Contributed by Mario Carneiro, 5-Sep-2016.)

Assertion

Ref	Expression
sadcom	⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐴 sadd 𝐵) = (𝐵 sadd 𝐴))

Proof of Theorem sadcom

Dummy variables 𝑘 𝑐 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hadcoma 1600	. . . 4 ⊢ (hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)) ↔ hadd(𝑘 ∈ 𝐵, 𝑘 ∈ 𝐴, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)))
2	1	a1i 11	. . 3 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘)) ↔ hadd(𝑘 ∈ 𝐵, 𝑘 ∈ 𝐴, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))))
3	2	rabbidv 3404	. 2 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))} = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐵, 𝑘 ∈ 𝐴, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
4		simpl 482	. . 3 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → 𝐴 ⊆ ℕ0)
5		simpr 484	. . 3 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → 𝐵 ⊆ ℕ0)
6		eqid 2734	. . 3 ⊢ seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 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))))
7	4, 5, 6	sadfval 16377	. 2 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
8		cadcoma 1613	. . . . . . 7 ⊢ (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐) ↔ cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐴, ∅ ∈ 𝑐))
9	8	a1i 11	. . . . . 6 ⊢ ((𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐) ↔ cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐴, ∅ ∈ 𝑐)))
10	9	ifbid 4501	. . . . 5 ⊢ ((𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0) → if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐴, ∅ ∈ 𝑐), 1o, ∅))
11	10	mpoeq3ia 7434	. . . 4 ⊢ (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)) = (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐴, ∅ ∈ 𝑐), 1o, ∅))
12		seqeq2 13926	. . . 4 ⊢ ((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)) = (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐴, ∅ ∈ 𝑐), 1o, ∅)) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐴, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))))
13	11, 12	ax-mp 5	. . 3 ⊢ seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐵, 𝑚 ∈ 𝐴, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
14	5, 4, 13	sadfval 16377	. 2 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐵 sadd 𝐴) = {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝐵, 𝑘 ∈ 𝐴, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
15	3, 7, 14	3eqtr4d 2779	1 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐴 sadd 𝐵) = (𝐵 sadd 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  haddwhad 1594  caddwcad 1607   ∈ wcel 2113  {crab 3397   ⊆ wss 3899  ∅c0 4283  ifcif 4477   ↦ cmpt 5177  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  1_oc1o 8388  2_oc2o 8389  0cc0 11024  1c1 11025   − cmin 11362  ℕ₀cn0 12399  seqcseq 13922   sadd csad 16345

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-1cn 11082  ax-addcl 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-had 1595  df-cad 1608  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12144  df-n0 12400  df-seq 13923  df-sad 16376

This theorem is referenced by:  sadid2  16394