hashiun - Intuitionistic Logic Explorer

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Theorem hashiun 11643

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 8042	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 1 ∈ ℂ)
5	1, 2, 3, 4	fsumiun 11642	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1)
6	2	ralrimiva 2570	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin)
7		iunfidisj 7012	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
8	1, 6, 3, 7	syl3anc 1249	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)
9		ax-1cn 7972	. . . 4 ⊢ 1 ∈ ℂ
10		fsumconst 11619	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1))
11	8, 9, 10	sylancl 413	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1))
12		hashcl 10873	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ₀)
13		nn0cn 9259	. . . 4 ⊢ ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℕ₀ → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ)
14		mulrid 8023	. . . 4 ⊢ ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) ∈ ℂ → ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1) = (♯‘∪ 𝑥 ∈ 𝐴 𝐵))
15	8, 12, 13, 14	4syl 18	. . 3 ⊢ (𝜑 → ((♯‘∪ 𝑥 ∈ 𝐴 𝐵) · 1) = (♯‘∪ 𝑥 ∈ 𝐴 𝐵))
16	11, 15	eqtrd 2229	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵1 = (♯‘∪ 𝑥 ∈ 𝐴 𝐵))
17		fsumconst 11619	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1))
18	2, 9, 17	sylancl 413	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = ((♯‘𝐵) · 1))
19		hashcl 10873	. . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
20		nn0cn 9259	. . . . 5 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
21		mulrid 8023	. . . . 5 ⊢ ((♯‘𝐵) ∈ ℂ → ((♯‘𝐵) · 1) = (♯‘𝐵))
22	2, 19, 20, 21	4syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((♯‘𝐵) · 1) = (♯‘𝐵))
23	18, 22	eqtrd 2229	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 1 = (♯‘𝐵))
24	23	sumeq2dv 11533	. 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 1 = Σ𝑥 ∈ 𝐴 (♯‘𝐵))
25	5, 16, 24	3eqtr3d 2237	1 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ𝑥 ∈ 𝐴 (♯‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2167 ∀wral 2475 ∪ ciun 3916 Disj wdisj 4010 ‘cfv 5258 (class class class)co 5922 Fincfn 6799 ℂcc 7877 1c1 7880 · cmul 7884 ℕ₀cn0 9249 ♯chash 10867 Σcsu 11518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-disj 4011 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-seqfrec 10540 df-exp 10631 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-sumdc 11519

This theorem is referenced by: hash2iun 11644 hashrabrex 11646 hashuni 11647 phisum 12409 lgsquadlem1 15318 lgsquadlem2 15319

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