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Mirrors > Home > MPE Home > Th. List > hashiun | Structured version Visualization version GIF version |
Description: The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
fsumiun.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumiun.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) |
fsumiun.3 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) |
Ref | Expression |
---|---|
hashiun | ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumiun.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsumiun.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) | |
3 | fsumiun.3 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
4 | 1cnd 10358 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 1 ∈ ℂ) | |
5 | 1, 2, 3, 4 | fsumiun 14934 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1) |
6 | 2 | ralrimiva 3175 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) |
7 | iunfi 8529 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin) | |
8 | 1, 6, 7 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin) |
9 | ax-1cn 10317 | . . . 4 ⊢ 1 ∈ ℂ | |
10 | fsumconst 14903 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1)) | |
11 | 8, 9, 10 | sylancl 580 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1)) |
12 | hashcl 13444 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ0) | |
13 | nn0cn 11636 | . . . 4 ⊢ ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ0 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ) | |
14 | mulid1 10361 | . . . 4 ⊢ ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ → ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1) = (♯‘∪ 𝑥 ∈ 𝐴 𝐵)) | |
15 | 8, 12, 13, 14 | 4syl 19 | . . 3 ⊢ (𝜑 → ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1) = (♯‘∪ 𝑥 ∈ 𝐴 𝐵)) |
16 | 11, 15 | eqtrd 2861 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = (♯‘∪ 𝑥 ∈ 𝐴 𝐵)) |
17 | fsumconst 14903 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) | |
18 | 2, 9, 17 | sylancl 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) |
19 | hashcl 13444 | . . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
20 | nn0cn 11636 | . . . . 5 ⊢ ((♯‘𝐵) ∈ ℕ0 → (♯‘𝐵) ∈ ℂ) | |
21 | mulid1 10361 | . . . . 5 ⊢ ((♯‘𝐵) ∈ ℂ → ((♯‘𝐵) · 1) = (♯‘𝐵)) | |
22 | 2, 19, 20, 21 | 4syl 19 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((♯‘𝐵) · 1) = (♯‘𝐵)) |
23 | 18, 22 | eqtrd 2861 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = (♯‘𝐵)) |
24 | 23 | sumeq2dv 14817 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1 = Σ𝑥 ∈ 𝐴 (♯‘𝐵)) |
25 | 5, 16, 24 | 3eqtr3d 2869 | 1 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∪ ciun 4742 Disj wdisj 4843 ‘cfv 6127 (class class class)co 6910 Fincfn 8228 ℂcc 10257 1c1 10260 · cmul 10264 ℕ0cn0 11625 ♯chash 13417 Σcsu 14800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-disj 4844 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-sum 14801 |
This theorem is referenced by: hash2iun 14936 hashrabrex 14938 hashuni 14939 ackbijnn 14941 phisum 15873 cshwshashnsame 16183 lgsquadlem1 25525 lgsquadlem2 25526 numedglnl 26450 fusgreghash2wsp 27715 numclwwlk4 27797 hashunif 30105 poimirlem26 33978 poimirlem27 33979 |
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