Theorem hashreprin 34914

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 34910	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6		inss2 4189	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
8	1, 2, 3, 7	reprss 34911	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
9	5, 8	ssfid 9213	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10		1cnd 11175	. . 3 ⊢ (𝜑 → 1 ∈ ℂ)
11		fsumconst 15817	. . 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 593	. 2 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1))
13	10	ralrimivw 3158	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ)
14	5	olcd 885	. . . 4 ⊢ (𝜑 → ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ≥‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin))
15		sumss2 15753	. . . 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 848	. . 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 34912	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
18		incom 4161	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
19	18	oveq1i 7406	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) = ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)
20	19	eleq2i 2854	. . . . . . . . . 10 ⊢ (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀))
21	20	bibi1i 340	. . . . . . . . 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 338	. . . . . . . 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 232	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
24	23	baibd 547	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ ran 𝑐 ⊆ 𝐴))
25	24	ifbid 4504	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐 ⊆ 𝐴, 1, 0))
26		nnex 12216	. . . . . . . . 9 ⊢ ℕ ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
28	27	ralrimivw 3158	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
29	28	r19.21bi 3254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30		fzofi 13987	. . . . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
34	1	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐵 ⊆ ℕ)
35	2	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
36	3	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
37		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐 ∈ (𝐵(repr‘𝑆)𝑀))
38	34, 35, 36, 37	reprf 34906	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
39	38, 34	fssd 6709	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
40	29, 31, 33, 39	prodindf 33040	. . . . 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 2800	. . . 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 15729	. . 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 2797	. 2 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
44		hashcl 14369	. . . . 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 12544	. . 3 ⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
47	46	mulridd 11199	. 2 ⊢ (𝜑 → ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1) = (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)))
48	12, 43, 47	3eqtr3rd 2806	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 208   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ifcif 4480  ran crn 5648  ‘cfv 6521  (class class class)co 7396  Fincfn 8927  ℂcc 11071  0cc0 11073  1c1 11074   · cmul 11078  𝟭cind 12195  ℕcn 12210  ℕ₀cn0 12481  ℤcz 12568  ℤ_≥cuz 12839  ..^cfzo 13659  ♯chash 14343  Σcsu 15713  ∏cprod 15933  reprcrepr 34902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-map 8810  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-ind 12196  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-rp 12994  df-fz 13513  df-fzo 13660  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-sum 15714  df-prod 15934  df-repr 34903

This theorem is referenced by:  hashrepr  34919  breprexpnat  34928