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Theorem hashiun 15768

Description: The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)

**Hypotheses**
Ref	Expression
fsumiun.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumiun.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
fsumiun.3	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)

**Assertion**
Ref	Expression
hashiun	⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem hashiun Dummy variable 𝑘 is distinct from all other variables.

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 11209	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 1 ∈ ℂ)
5	1, 2, 3, 4	fsumiun 15767	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1)
6	2	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
7		iunfi 9340	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
8	1, 6, 7	syl2anc 585	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
9		ax-1cn 11168	. . . 4 ⊢ 1 ∈ ℂ
10		fsumconst 15736	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1))
11	8, 9, 10	sylancl 587	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1))
12		hashcl 14316	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ₀)
13		nn0cn 12482	. . . 4 ⊢ ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ₀ → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ)
14		mulrid 11212	. . . 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 2773	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = (♯‘∪ 𝑥 ∈ 𝐴 𝐵))
17		fsumconst 15736	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1))
18	2, 9, 17	sylancl 587	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1))
19		hashcl 14316	. . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
20		nn0cn 12482	. . . . 5 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
21		mulrid 11212	. . . . 5 ⊢ ((♯‘𝐵) ∈ ℂ → ((♯‘𝐵) · 1) = (♯‘𝐵))
22	2, 19, 20, 21	4syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((♯‘𝐵) · 1) = (♯‘𝐵))
23	18, 22	eqtrd 2773	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = (♯‘𝐵))
24	23	sumeq2dv 15649	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1 = Σ𝑥 ∈ 𝐴 (♯‘𝐵))
25	5, 16, 24	3eqtr3d 2781	1 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∪ ciun 4998 Disj wdisj 5114 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 1c1 11111 · cmul 11115 ℕ₀cn0 12472 ♯chash 14290 Σcsu 15632

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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633

This theorem is referenced by: hash2iun 15769 hashrabrex 15771 hashuni 15772 ackbijnn 15774 phisum 16723 cshwshashnsame 17037 lgsquadlem1 26883 lgsquadlem2 26884 numedglnl 28404 fusgreghash2wsp 29591 numclwwlk4 29639 hashunif 32018 poimirlem26 36514 poimirlem27 36515

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