sumhashdc - Intuitionistic Logic Explorer

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Theorem sumhashdc 12856

Description: The sum of 1 over a set is the size of the set. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 20-May-2014.)

**Assertion**
Ref	Expression
sumhashdc	⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 1, 0) = (♯‘𝐴))

Distinct variable groups: 𝑥,𝑘,𝐴 𝐵,𝑘,𝑥

Proof of Theorem sumhashdc Dummy variable 𝐶 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssfidc 7087	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → 𝐴 ∈ Fin)
2		ax-1cn 8080	. . 3 ⊢ 1 ∈ ℂ
3		fsumconst 11951	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑘 ∈ 𝐴 1 = ((♯‘𝐴) · 1))
4	1, 2, 3	sylancl 413	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐴 1 = ((♯‘𝐴) · 1))
5		simp2 1022	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
6		simp3 1023	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
7	2	rgenw 2585	. . . 4 ⊢ ∀𝑘 ∈ 𝐴 1 ∈ ℂ
8	7	a1i 9	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 1 ∈ ℂ)
9		simp1 1021	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
10	9	olcd 739	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → ((𝐶 ∈ ℤ ∧ 𝐵 ⊆ (ℤ_≥‘𝐶) ∧ ∀𝑥 ∈ (ℤ_≥‘𝐶)DECID 𝑥 ∈ 𝐵) ∨ 𝐵 ∈ Fin))
11	5, 6, 8, 10	isumss2 11890	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐴 1 = Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 1, 0))
12		hashcl 10990	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
13	1, 12	syl 14	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → (♯‘𝐴) ∈ ℕ₀)
14	13	nn0cnd 9412	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → (♯‘𝐴) ∈ ℂ)
15	14	mulridd 8151	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → ((♯‘𝐴) · 1) = (♯‘𝐴))
16	4, 11, 15	3eqtr3d 2270	1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 1, 0) = (♯‘𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 DECID wdc 839 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ⊆ wss 3197 ifcif 3602 ‘cfv 5314 (class class class)co 5994 Fincfn 6877 ℂcc 7985 0cc0 7987 1c1 7988 · cmul 7992 ℕ₀cn0 9357 ℤcz 9434 ℤ_≥cuz 9710 ♯chash 10984 Σcsu 11850

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-iinf 4677 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-mulrcl 8086 ax-addcom 8087 ax-mulcom 8088 ax-addass 8089 ax-mulass 8090 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-1rid 8094 ax-0id 8095 ax-rnegex 8096 ax-precex 8097 ax-cnre 8098 ax-pre-ltirr 8099 ax-pre-ltwlin 8100 ax-pre-lttrn 8101 ax-pre-apti 8102 ax-pre-ltadd 8103 ax-pre-mulgt0 8104 ax-pre-mulext 8105 ax-arch 8106 ax-caucvg 8107

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4381 df-po 4384 df-iso 4385 df-iord 4454 df-on 4456 df-ilim 4457 df-suc 4459 df-iom 4680 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-rn 4727 df-res 4728 df-ima 4729 df-iota 5274 df-fun 5316 df-fn 5317 df-f 5318 df-f1 5319 df-fo 5320 df-f1o 5321 df-fv 5322 df-isom 5323 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-1st 6276 df-2nd 6277 df-recs 6441 df-irdg 6506 df-frec 6527 df-1o 6552 df-oadd 6556 df-er 6670 df-en 6878 df-dom 6879 df-fin 6880 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-sub 8307 df-neg 8308 df-reap 8710 df-ap 8717 df-div 8808 df-inn 9099 df-2 9157 df-3 9158 df-4 9159 df-n0 9358 df-z 9435 df-uz 9711 df-q 9803 df-rp 9838 df-fz 10193 df-fzo 10327 df-seqfrec 10657 df-exp 10748 df-ihash 10985 df-cj 11339 df-re 11340 df-im 11341 df-rsqrt 11495 df-abs 11496 df-clim 11776 df-sumdc 11851

This theorem is referenced by: pcfac 12859

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