Theorem hashreprin 34576

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 34572	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6		inss2 4220	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
8	1, 2, 3, 7	reprss 34573	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
9	5, 8	ssfid 9284	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10		1cnd 11239	. . 3 ⊢ (𝜑 → 1 ∈ ℂ)
11		fsumconst 15809	. . 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 584	. 2 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1))
13	10	ralrimivw 3137	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ)
14	5	olcd 874	. . . 4 ⊢ (𝜑 → ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ≥‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin))
15		sumss2 15745	. . . 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 837	. . 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 34574	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
18		incom 4191	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
19	18	oveq1i 7424	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) = ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)
20	19	eleq2i 2825	. . . . . . . . . 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 336	. . . . . . . 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 230	. . . . . . 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 4531	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐 ⊆ 𝐴, 1, 0))
26		nnex 12255	. . . . . . . . 9 ⊢ ℕ ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
28	27	ralrimivw 3137	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
29	28	r19.21bi 3238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30		fzofi 13998	. . . . . . 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 34568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
39	38, 34	fssd 6734	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
40	29, 31, 33, 39	prodindf 32795	. . . . 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 2772	. . . 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 15721	. . 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 2769	. 2 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
44		hashcl 14378	. . . . 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 12573	. . 3 ⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
47	46	mulridd 11261	. 2 ⊢ (𝜑 → ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1) = (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)))
48	12, 43, 47	3eqtr3rd 2778	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 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∀wral 3050  Vcvv 3464   ∩ cin 3932   ⊆ wss 3933  ifcif 4507  ran crn 5668  ‘cfv 6542  (class class class)co 7414  Fincfn 8968  ℂcc 11136  0cc0 11138  1c1 11139   · cmul 11143  ℕcn 12249  ℕ₀cn0 12510  ℤcz 12597  ℤ_≥cuz 12861  ..^cfzo 13677  ♯chash 14352  Σcsu 15705  ∏cprod 15922  𝟭cind 32782  reprcrepr 34564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-int 4929  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-1st 7997  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-1o 8489  df-er 8728  df-map 8851  df-pm 8852  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-sup 9465  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11477  df-neg 11478  df-div 11904  df-nn 12250  df-2 12312  df-3 12313  df-n0 12511  df-z 12598  df-uz 12862  df-rp 13018  df-fz 13531  df-fzo 13678  df-seq 14026  df-exp 14086  df-hash 14353  df-cj 15121  df-re 15122  df-im 15123  df-sqrt 15257  df-abs 15258  df-clim 15507  df-sum 15706  df-prod 15923  df-ind 32783  df-repr 34565

This theorem is referenced by:  hashrepr  34581  breprexpnat  34590