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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 10630 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 1 ∈ ℂ) | |
5 | 1, 2, 3, 4 | fsumiun 15170 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1) |
6 | 2 | ralrimiva 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) |
7 | iunfi 8806 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin) | |
8 | 1, 6, 7 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin) |
9 | ax-1cn 10589 | . . . 4 ⊢ 1 ∈ ℂ | |
10 | fsumconst 15139 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1)) | |
11 | 8, 9, 10 | sylancl 588 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1)) |
12 | hashcl 13711 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ0) | |
13 | nn0cn 11901 | . . . 4 ⊢ ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ0 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ) | |
14 | mulid1 10633 | . . . 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 2856 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = (♯‘∪ 𝑥 ∈ 𝐴 𝐵)) |
17 | fsumconst 15139 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) | |
18 | 2, 9, 17 | sylancl 588 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) |
19 | hashcl 13711 | . . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
20 | nn0cn 11901 | . . . . 5 ⊢ ((♯‘𝐵) ∈ ℕ0 → (♯‘𝐵) ∈ ℂ) | |
21 | mulid1 10633 | . . . . 5 ⊢ ((♯‘𝐵) ∈ ℂ → ((♯‘𝐵) · 1) = (♯‘𝐵)) | |
22 | 2, 19, 20, 21 | 4syl 19 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((♯‘𝐵) · 1) = (♯‘𝐵)) |
23 | 18, 22 | eqtrd 2856 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = (♯‘𝐵)) |
24 | 23 | sumeq2dv 15054 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1 = Σ𝑥 ∈ 𝐴 (♯‘𝐵)) |
25 | 5, 16, 24 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∪ ciun 4911 Disj wdisj 5023 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 ℂcc 10529 1c1 10532 · cmul 10536 ℕ0cn0 11891 ♯chash 13684 Σcsu 15036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 |
This theorem is referenced by: hash2iun 15172 hashrabrex 15174 hashuni 15175 ackbijnn 15177 phisum 16121 cshwshashnsame 16431 lgsquadlem1 25950 lgsquadlem2 25951 numedglnl 26923 fusgreghash2wsp 28111 numclwwlk4 28159 hashunif 30522 poimirlem26 34912 poimirlem27 34913 |
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