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Theorem hashiun 15731

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 11114	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 1 ∈ ℂ)
5	1, 2, 3, 4	fsumiun 15730	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1)
6	2	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
7		iunfi 9234	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
8	1, 6, 7	syl2anc 584	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
9		ax-1cn 11071	. . . 4 ⊢ 1 ∈ ℂ
10		fsumconst 15699	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1))
11	8, 9, 10	sylancl 586	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1))
12		hashcl 14265	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ₀)
13		nn0cn 12398	. . . 4 ⊢ ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ₀ → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ)
14		mulrid 11117	. . . 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 2768	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = (♯‘∪ 𝑥 ∈ 𝐴 𝐵))
17		fsumconst 15699	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1))
18	2, 9, 17	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1))
19		hashcl 14265	. . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
20		nn0cn 12398	. . . . 5 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
21		mulrid 11117	. . . . 5 ⊢ ((♯‘𝐵) ∈ ℂ → ((♯‘𝐵) · 1) = (♯‘𝐵))
22	2, 19, 20, 21	4syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((♯‘𝐵) · 1) = (♯‘𝐵))
23	18, 22	eqtrd 2768	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = (♯‘𝐵))
24	23	sumeq2dv 15611	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1 = Σ𝑥 ∈ 𝐴 (♯‘𝐵))
25	5, 16, 24	3eqtr3d 2776	1 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ∪ ciun 4941 Disj wdisj 5060 ‘cfv 6486 (class class class)co 7352 Fincfn 8875 ℂcc 11011 1c1 11014 · cmul 11018 ℕ₀cn0 12388 ♯chash 14239 Σcsu 15595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-disj 5061 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-clim 15397 df-sum 15596

This theorem is referenced by: hash2iun 15732 hashrabrex 15734 hashuni 15735 ackbijnn 15737 phisum 16704 cshwshashnsame 17017 lgsquadlem1 27319 lgsquadlem2 27320 numedglnl 29124 fusgreghash2wsp 30320 numclwwlk4 30368 hashunif 32793 poimirlem26 37706 poimirlem27 37707 grpods 42307

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