sumhashdc - Intuitionistic Logic Explorer

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Theorem sumhashdc 13041

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 7197	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → 𝐴 ∈ Fin)
2		ax-1cn 8219	. . 3 ⊢ 1 ∈ ℂ
3		fsumconst 12136	. . 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 1025	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
6		simp3 1026	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
7	2	rgenw 2597	. . . 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 1024	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin)
10	9	olcd 742	. . 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 12075	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → Σ𝑘 ∈ 𝐴 1 = Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 1, 0))
12		hashcl 11142	. . . . 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 9554	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → (♯‘𝐴) ∈ ℂ)
15	14	mulridd 8290	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴) → ((♯‘𝐴) · 1) = (♯‘𝐴))
16	4, 11, 15	3eqtr3d 2273	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 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 ∀wral 2520 ⊆ wss 3210 ifcif 3619 ‘cfv 5351 (class class class)co 6049 Fincfn 6974 ℂcc 8124 0cc0 8126 1c1 8127 · cmul 8131 ℕ₀cn0 9495 ℤcz 9576 ℤ_≥cuz 9852 ♯chash 11136 Σcsu 12034

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244 ax-arch 8245 ax-caucvg 8246

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-frec 6621 df-1o 6646 df-oadd 6650 df-er 6766 df-en 6975 df-dom 6976 df-fin 6977 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-n0 9496 df-z 9577 df-uz 9853 df-q 9951 df-rp 9986 df-fz 10342 df-fzo 10476 df-seqfrec 10809 df-exp 10900 df-ihash 11137 df-cj 11523 df-re 11524 df-im 11525 df-rsqrt 11679 df-abs 11680 df-clim 11960 df-sumdc 12035

This theorem is referenced by: pcfac 13044

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