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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 16407 | . . . . 5 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐴 sadd 𝐵) ⊆ ℕ0) | |
4 | 1, 2, 3 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐴 sadd 𝐵) ⊆ ℕ0) |
5 | 4 | sseld 3976 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) → 𝑘 ∈ ℕ0)) |
6 | 1, 2 | unssd 4181 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℕ0) |
7 | 6 | sseld 3976 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) → 𝑘 ∈ ℕ0)) |
8 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ⊆ ℕ0) |
9 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ⊆ ℕ0) |
10 | saddisj.3 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∩ 𝐵) = ∅) |
12 | eqid 2726 | . . . . 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 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
14 | 8, 9, 11, 12, 13 | saddisjlem 16409 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
15 | 14 | ex 412 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))) |
16 | 5, 7, 15 | pm5.21ndd 379 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
17 | 16 | eqrdv 2724 | 1 ⊢ (𝜑 → (𝐴 sadd 𝐵) = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 caddwcad 1599 ∈ wcel 2098 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 ifcif 4523 ↦ cmpt 5224 (class class class)co 7404 ∈ cmpo 7406 1oc1o 8457 2oc2o 8458 0cc0 11109 1c1 11110 − cmin 11445 ℕ0cn0 12473 seqcseq 13969 sadd csad 16365 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-seq 13970 df-sad 16396 |
This theorem is referenced by: sadid1 16413 bitsres 16418 |
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