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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 10056 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 1 ∈ ℂ) | |
5 | 1, 2, 3, 4 | fsumiun 14553 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1) |
6 | 2 | ralrimiva 2966 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) |
7 | iunfi 8254 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin) | |
8 | 1, 6, 7 | syl2anc 693 | . . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin) |
9 | ax-1cn 9994 | . . . 4 ⊢ 1 ∈ ℂ | |
10 | fsumconst 14522 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((#‘∪ 𝑥 ∈ 𝐴 𝐵) · 1)) | |
11 | 8, 9, 10 | sylancl 694 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((#‘∪ 𝑥 ∈ 𝐴 𝐵) · 1)) |
12 | hashcl 13147 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin → (#‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ0) | |
13 | nn0cn 11302 | . . . 4 ⊢ ((#‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ0 → (#‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ) | |
14 | mulid1 10037 | . . . 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 2656 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = (#‘∪ 𝑥 ∈ 𝐴 𝐵)) |
17 | fsumconst 14522 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ 𝐵 1 = ((#‘𝐵) · 1)) | |
18 | 2, 9, 17 | sylancl 694 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = ((#‘𝐵) · 1)) |
19 | hashcl 13147 | . . . . 5 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0) | |
20 | nn0cn 11302 | . . . . 5 ⊢ ((#‘𝐵) ∈ ℕ0 → (#‘𝐵) ∈ ℂ) | |
21 | mulid1 10037 | . . . . 5 ⊢ ((#‘𝐵) ∈ ℂ → ((#‘𝐵) · 1) = (#‘𝐵)) | |
22 | 2, 19, 20, 21 | 4syl 19 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((#‘𝐵) · 1) = (#‘𝐵)) |
23 | 18, 22 | eqtrd 2656 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = (#‘𝐵)) |
24 | 23 | sumeq2dv 14433 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1 = Σ𝑥 ∈ 𝐴 (#‘𝐵)) |
25 | 5, 16, 24 | 3eqtr3d 2664 | 1 ⊢ (𝜑 → (#‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (#‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∪ ciun 4520 Disj wdisj 4620 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 1c1 9937 · cmul 9941 ℕ0cn0 11292 #chash 13117 Σcsu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 |
This theorem is referenced by: hash2iun 14555 hashrabrex 14557 hashuni 14558 ackbijnn 14560 phisum 15495 cshwshashnsame 15810 lgsquadlem1 25105 lgsquadlem2 25106 numedglnl 26039 fusgreghash2wsp 27202 numclwwlk4 27244 hashunif 29562 poimirlem26 33435 poimirlem27 33436 |
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