Theorem hashreprin 33927

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 33923	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6		inss2 4230	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
8	1, 2, 3, 7	reprss 33924	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
9	5, 8	ssfid 9270	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10		1cnd 11214	. . 3 ⊢ (𝜑 → 1 ∈ ℂ)
11		fsumconst 15741	. . 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 3149	. . . 4 ⊢ (𝜑 → ∀𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ)
14	5	olcd 871	. . . 4 ⊢ (𝜑 → ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ≥‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin))
15		sumss2 15677	. . . 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 835	. . 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 33925	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
18		incom 4202	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
19	18	oveq1i 7422	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) = ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)
20	19	eleq2i 2824	. . . . . . . . . 10 ⊢ (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀))
21	20	bibi1i 337	. . . . . . . . 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 4552	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐 ⊆ 𝐴, 1, 0))
26		nnex 12223	. . . . . . . . 9 ⊢ ℕ ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
28	27	ralrimivw 3149	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
29	28	r19.21bi 3247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30		fzofi 13944	. . . . . . 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 33919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
39	38, 34	fssd 6736	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
40	29, 31, 33, 39	prodindf 33316	. . . . 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 2774	. . . 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 15654	. . 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 2771	. 2 ⊢ (𝜑 → Σ𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎)))
44		hashcl 14321	. . . . 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 12539	. . 3 ⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
47	46	mulridd 11236	. 2 ⊢ (𝜑 → ((♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) · 1) = (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)))
48	12, 43, 47	3eqtr3rd 2780	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 844   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ∩ cin 3948   ⊆ wss 3949  ifcif 4529  ran crn 5678  ‘cfv 6544  (class class class)co 7412  Fincfn 8942  ℂcc 11111  0cc0 11113  1c1 11114   · cmul 11118  ℕcn 12217  ℕ₀cn0 12477  ℤcz 12563  ℤ_≥cuz 12827  ..^cfzo 13632  ♯chash 14295  Σcsu 15637  ∏cprod 15854  𝟭cind 33303  reprcrepr 33915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9440  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-clim 15437  df-sum 15638  df-prod 15855  df-ind 33304  df-repr 33916

This theorem is referenced by:  hashrepr  33932  breprexpnat  33941