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Theorem hashreprin 32005
 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 (𝜑𝐴 ⊆ ℕ)
reprval.m (𝜑𝑀 ∈ ℤ)
reprval.s (𝜑𝑆 ∈ ℕ0)
hashreprin.b (𝜑𝐵 ∈ Fin)
hashreprin.1 (𝜑𝐵 ⊆ ℕ)
Assertion
Ref Expression
hashreprin (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
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
StepHypRef Expression
1 hashreprin.1 . . . . 5 (𝜑𝐵 ⊆ ℕ)
2 reprval.m . . . . 5 (𝜑𝑀 ∈ ℤ)
3 reprval.s . . . . 5 (𝜑𝑆 ∈ ℕ0)
4 hashreprin.b . . . . 5 (𝜑𝐵 ∈ Fin)
51, 2, 3, 4reprfi 32001 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6 inss2 4159 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
76a1i 11 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ 𝐵)
81, 2, 3, 7reprss 32002 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
95, 8ssfid 8729 . . 3 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10 1cnd 10629 . . 3 (𝜑 → 1 ∈ ℂ)
11 fsumconst 15141 . . 3 ((((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1))
129, 10, 11syl2anc 587 . 2 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1))
1310ralrimivw 3153 . . . 4 (𝜑 → ∀𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ)
145olcd 871 . . . 4 (𝜑 → ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin))
15 sumss2 15079 . . . 4 (((((𝐴𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀) ∧ ∀𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ) ∧ ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin)) → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0))
168, 13, 14, 15syl21anc 836 . . 3 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0))
171, 2, 3reprinrn 32003 . . . . . . . 8 (𝜑 → (𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)))
18 incom 4131 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
1918oveq1i 7149 . . . . . . . . . . 11 ((𝐵𝐴)(repr‘𝑆)𝑀) = ((𝐴𝐵)(repr‘𝑆)𝑀)
2019eleq2i 2884 . . . . . . . . . 10 (𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀))
2120bibi1i 342 . . . . . . . . 9 ((𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)) ↔ (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)))
2221imbi2i 339 . . . . . . . 8 ((𝜑 → (𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴))) ↔ (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴))))
2317, 22mpbi 233 . . . . . . 7 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)))
2423baibd 543 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ ran 𝑐𝐴))
2524ifbid 4450 . . . . 5 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐𝐴, 1, 0))
26 nnex 11635 . . . . . . . . 9 ℕ ∈ V
2726a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
2827ralrimivw 3153 . . . . . . 7 (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
2928r19.21bi 3176 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30 fzofi 13341 . . . . . . 7 (0..^𝑆) ∈ Fin
3130a1i 11 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (0..^𝑆) ∈ Fin)
32 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
3332adantr 484 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
341adantr 484 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐵 ⊆ ℕ)
352adantr 484 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
363adantr 484 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
37 simpr 488 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐 ∈ (𝐵(repr‘𝑆)𝑀))
3834, 35, 36, 37reprf 31997 . . . . . . 7 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
3938, 34fssd 6506 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
4029, 31, 33, 39prodindf 31396 . . . . 5 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) = if(ran 𝑐𝐴, 1, 0))
4125, 40eqtr4d 2839 . . . 4 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
4241sumeq2dv 15056 . . 3 (𝜑 → Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
4316, 42eqtrd 2836 . 2 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
44 hashcl 13717 . . . . 5 (((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
459, 44syl 17 . . . 4 (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
4645nn0cnd 11949 . . 3 (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
4746mulid1d 10651 . 2 (𝜑 → ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1) = (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)))
4812, 43, 473eqtr3rd 2845 1 (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112  ∀wral 3109  Vcvv 3444   ∩ cin 3883   ⊆ wss 3884  ifcif 4428  ran crn 5524  ‘cfv 6328  (class class class)co 7139  Fincfn 8496  ℂcc 10528  0cc0 10530  1c1 10531   · cmul 10535  ℕcn 11629  ℕ0cn0 11889  ℤcz 11973  ℤ≥cuz 12235  ..^cfzo 13032  ♯chash 13690  Σcsu 15038  ∏cprod 15255  𝟭cind 31383  reprcrepr 31993 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-sum 15039  df-prod 15256  df-ind 31384  df-repr 31994 This theorem is referenced by:  hashrepr  32010  breprexpnat  32019
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