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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 16347 | . . . . 5 ⊢ ((𝐴 ⊆ ℕ0 ∧ 𝐵 ⊆ ℕ0) → (𝐴 sadd 𝐵) ⊆ ℕ0) | |
4 | 1, 2, 3 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝐴 sadd 𝐵) ⊆ ℕ0) |
5 | 4 | sseld 3944 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) → 𝑘 ∈ ℕ0)) |
6 | 1, 2 | unssd 4147 | . . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ℕ0) |
7 | 6 | sseld 3944 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 ∪ 𝐵) → 𝑘 ∈ ℕ0)) |
8 | 1 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ⊆ ℕ0) |
9 | 2 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ⊆ ℕ0) |
10 | saddisj.3 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
11 | 10 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 ∩ 𝐵) = ∅) |
12 | eqid 2733 | . . . . 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 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
14 | 8, 9, 11, 12, 13 | saddisjlem 16349 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
15 | 14 | ex 414 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵)))) |
16 | 5, 7, 15 | pm5.21ndd 381 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 sadd 𝐵) ↔ 𝑘 ∈ (𝐴 ∪ 𝐵))) |
17 | 16 | eqrdv 2731 | 1 ⊢ (𝜑 → (𝐴 sadd 𝐵) = (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 caddwcad 1608 ∈ wcel 2107 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4283 ifcif 4487 ↦ cmpt 5189 (class class class)co 7358 ∈ cmpo 7360 1oc1o 8406 2oc2o 8407 0cc0 11056 1c1 11057 − cmin 11390 ℕ0cn0 12418 seqcseq 13912 sadd csad 16305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-had 1596 df-cad 1609 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-seq 13913 df-sad 16336 |
This theorem is referenced by: sadid1 16353 bitsres 16358 |
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