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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 15813 | . . . . 5 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐴 sadd 𝐵) ⊆ ℕ0) | |
4 | 1, 2, 3 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐴 sadd 𝐵) ⊆ ℕ0) |
5 | 4 | sseld 3968 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) → 𝑘 ∈ ℕ0)) |
6 | 1, 2 | unssd 4164 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℕ0) |
7 | 6 | sseld 3968 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) → 𝑘 ∈ ℕ0)) |
8 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ⊆ ℕ0) |
9 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ⊆ ℕ0) |
10 | saddisj.3 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
11 | 10 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∩ 𝐵) = ∅) |
12 | eqid 2823 | . . . . 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 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
14 | 8, 9, 11, 12, 13 | saddisjlem 15815 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
15 | 14 | ex 415 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))) |
16 | 5, 7, 15 | pm5.21ndd 383 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
17 | 16 | eqrdv 2821 | 1 ⊢ (𝜑 → (𝐴 sadd 𝐵) = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 caddwcad 1607 ∈ wcel 2114 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 ifcif 4469 ↦ cmpt 5148 (class class class)co 7158 ∈ cmpo 7160 1oc1o 8097 2oc2o 8098 0cc0 10539 1c1 10540 − cmin 10872 ℕ0cn0 11900 seqcseq 13372 sadd csad 15771 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-had 1594 df-cad 1608 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-seq 13373 df-sad 15802 |
This theorem is referenced by: sadid1 15819 bitsres 15824 |
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