Theorem hashreprin 32500

Description: Express a sum of representations over an intersection using a product of the indicator function. (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
reprval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
reprval.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
reprval.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
hashreprin.b	⊢ (𝜑 → 𝐵 ∈ Fin)
hashreprin.1	⊢ (𝜑 → 𝐵 ⊆ ℕ)

Assertion

Ref	Expression
hashreprin	⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))

Distinct variable groups:   𝐴,𝑐   𝑀,𝑐   𝑆,𝑎,𝑐   𝜑,𝑐   𝐴,𝑎   𝐵,𝑎,𝑐   𝑀,𝑎   𝜑,𝑎

Proof of Theorem hashreprin

Step	Hyp	Ref	Expression
1		hashreprin.1	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
2		reprval.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		reprval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
4		hashreprin.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin)
5	1, 2, 3, 4	reprfi 32496	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6		inss2 4160	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
8	1, 2, 3, 7	reprss 32497	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
9	5, 8	ssfid 8971	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10		1cnd 10901	. . 3 ⊢ (𝜑 → 1 ∈ ℂ)
11		fsumconst 15430	. . 3 ⊢ ((((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1))
12	9, 10, 11	syl2anc 583	. 2 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1))
13	10	ralrimivw 3108	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ)
14	5	olcd 870	. . . 4 ⊢ (𝜑 → ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ≥‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin))
15		sumss2 15366	. . . 4 ⊢ (((((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀) ∧ ∀𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ) ∧ ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ≥‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin)) → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0))
16	8, 13, 14, 15	syl21anc 834	. . 3 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0))
17	1, 2, 3	reprinrn 32498	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
18		incom 4131	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
19	18	oveq1i 7265	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) = ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)
20	19	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀))
21	20	bibi1i 338	. . . . . . . . 9 ⊢ ((𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)) ↔ (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
22	21	imbi2i 335	. . . . . . . 8 ⊢ ((𝜑 → (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴))) ↔ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴))))
23	17, 22	mpbi 229	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
24	23	baibd 539	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ ran 𝑐 ⊆ 𝐴))
25	24	ifbid 4479	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐 ⊆ 𝐴, 1, 0))
26		nnex 11909	. . . . . . . . 9 ⊢ ℕ ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
28	27	ralrimivw 3108	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
29	28	r19.21bi 3132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30		fzofi 13622	. . . . . . 7 ⊢ (0..^𝑆) ∈ Fin
31	30	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (0..^𝑆) ∈ Fin)
32		reprval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
33	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
34	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐵 ⊆ ℕ)
35	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
36	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
37		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐 ∈ (𝐵(repr‘𝑆)𝑀))
38	34, 35, 36, 37	reprf 32492	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
39	38, 34	fssd 6602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
40	29, 31, 33, 39	prodindf 31891	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)) = if(ran 𝑐 ⊆ 𝐴, 1, 0))
41	25, 40	eqtr4d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
42	41	sumeq2dv 15343	. . 3 ⊢ (𝜑 → Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
43	16, 42	eqtrd 2778	. 2 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
44		hashcl 13999	. . . . 5 ⊢ (((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ∈ Fin → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
45	9, 44	syl 17	. . . 4 ⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
46	45	nn0cnd 12225	. . 3 ⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
47	46	mulid1d 10923	. 2 ⊢ (𝜑 → ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1) = (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)))
48	12, 43, 47	3eqtr3rd 2787	1 ⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ifcif 4456  ran crn 5581  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  ℂcc 10800  0cc0 10802  1c1 10803   · cmul 10807  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ..^cfzo 13311  ♯chash 13972  Σcsu 15325  ∏cprod 15543  𝟭cind 31878  reprcrepr 32488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-prod 15544  df-ind 31879  df-repr 32489

This theorem is referenced by:  hashrepr  32505  breprexpnat  32514