Theorem hashreprin 34726

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 34722	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6		inss2 4188	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
8	1, 2, 3, 7	reprss 34723	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
9	5, 8	ssfid 9167	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10		1cnd 11125	. . 3 ⊢ (𝜑 → 1 ∈ ℂ)
11		fsumconst 15711	. . 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 3130	. . . 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 15647	. . . 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 34724	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐴)))
18		incom 4159	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
19	18	oveq1i 7366	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ 𝐴)(repr‘𝑆)𝑀) = ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)
20	19	eleq2i 2826	. . . . . . . . . 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 4501	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐 ⊆ 𝐴, 1, 0))
26		nnex 12149	. . . . . . . . 9 ⊢ ℕ ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
28	27	ralrimivw 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
29	28	r19.21bi 3226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30		fzofi 13895	. . . . . . 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 34718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
39	38, 34	fssd 6677	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
40	29, 31, 33, 39	prodindf 32893	. . . . 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 15623	. . 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 14277	. . . . 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 12462	. . 3 ⊢ (𝜑 → (♯‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
47	46	mulridd 11147	. 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 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899  ifcif 4477  ran crn 5623  ‘cfv 6490  (class class class)co 7356  Fincfn 8881  ℂcc 11022  0cc0 11024  1c1 11025   · cmul 11029  ℕcn 12143  ℕ₀cn0 12399  ℤcz 12486  ℤ_≥cuz 12749  ..^cfzo 13568  ♯chash 14251  Σcsu 15607  ∏cprod 15824  𝟭cind 32878  reprcrepr 34714

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-fz 13422  df-fzo 13569  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-clim 15409  df-sum 15608  df-prod 15825  df-ind 32879  df-repr 34715

This theorem is referenced by:  hashrepr  34731  breprexpnat  34740