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Mirrors > Home > MPE Home > Th. List > saddisj | Structured version Visualization version GIF version |
Description: The sum of disjoint sequences is the union of the sequences. (In this case, there are no carried bits.) (Contributed by Mario Carneiro, 9-Sep-2016.) |
Ref | Expression |
---|---|
saddisj.1 | ⊢ (𝜑 → 𝐴 ⊆ ℕ0) |
saddisj.2 | ⊢ (𝜑 → 𝐵 ⊆ ℕ0) |
saddisj.3 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) |
Ref | Expression |
---|---|
saddisj | ⊢ (𝜑 → (𝐴 sadd 𝐵) = (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | saddisj.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℕ0) | |
2 | saddisj.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℕ0) | |
3 | sadcl 16444 | . . . . 5 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐴 sadd 𝐵) ⊆ ℕ0) | |
4 | 1, 2, 3 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐴 sadd 𝐵) ⊆ ℕ0) |
5 | 4 | sseld 3981 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) → 𝑘 ∈ ℕ0)) |
6 | 1, 2 | unssd 4188 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℕ0) |
7 | 6 | sseld 3981 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) → 𝑘 ∈ ℕ0)) |
8 | 1 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ⊆ ℕ0) |
9 | 2 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ⊆ ℕ0) |
10 | saddisj.3 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
11 | 10 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∩ 𝐵) = ∅) |
12 | eqid 2728 | . . . . 5 ⊢ seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑥 − 1)))) = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (𝑥 − 1)))) | |
13 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
14 | 8, 9, 11, 12, 13 | saddisjlem 16446 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
15 | 14 | ex 411 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))) |
16 | 5, 7, 15 | pm5.21ndd 378 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
17 | 16 | eqrdv 2726 | 1 ⊢ (𝜑 → (𝐴 sadd 𝐵) = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 caddwcad 1599 ∈ wcel 2098 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 ifcif 4532 ↦ cmpt 5235 (class class class)co 7426 ∈ cmpo 7428 1oc1o 8486 2oc2o 8487 0cc0 11146 1c1 11147 − cmin 11482 ℕ0cn0 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-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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-nel 3044 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-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-seq 14007 df-sad 16433 |
This theorem is referenced by: sadid1 16450 bitsres 16455 |
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