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Theorem hashreprin 31304
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 31300 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6 inss2 4054 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
76a1i 11 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ 𝐵)
81, 2, 3, 7reprss 31301 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
95, 8ssfid 8473 . . 3 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10 1cnd 10373 . . 3 (𝜑 → 1 ∈ ℂ)
11 fsumconst 14930 . . 3 ((((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1))
129, 10, 11syl2anc 579 . 2 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1))
1310ralrimivw 3149 . . . 4 (𝜑 → ∀𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 ∈ ℂ)
145olcd 863 . . . 4 (𝜑 → ((𝐵(repr‘𝑆)𝑀) ⊆ (ℤ‘0) ∨ (𝐵(repr‘𝑆)𝑀) ∈ Fin))
15 sumss2 14868 . . . 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 828 . . 3 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0))
171, 2, 3reprinrn 31302 . . . . . . . 8 (𝜑 → (𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)))
18 incom 4028 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
1918oveq1i 6934 . . . . . . . . . . 11 ((𝐵𝐴)(repr‘𝑆)𝑀) = ((𝐴𝐵)(repr‘𝑆)𝑀)
2019eleq2i 2851 . . . . . . . . . 10 (𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀))
2120bibi1i 330 . . . . . . . . 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 328 . . . . . . . 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 222 . . . . . . 7 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)))
2423baibd 535 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ ran 𝑐𝐴))
2524ifbid 4329 . . . . 5 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐𝐴, 1, 0))
26 nnex 11385 . . . . . . . . 9 ℕ ∈ V
2726a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
2827ralrimivw 3149 . . . . . . 7 (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
2928r19.21bi 3114 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30 fzofi 13096 . . . . . . 7 (0..^𝑆) ∈ Fin
3130a1i 11 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (0..^𝑆) ∈ Fin)
32 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
3332adantr 474 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
341adantr 474 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐵 ⊆ ℕ)
352adantr 474 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
363adantr 474 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
37 simpr 479 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐 ∈ (𝐵(repr‘𝑆)𝑀))
3834, 35, 36, 37reprf 31296 . . . . . . 7 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
3938, 34fssd 6307 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
4029, 31, 33, 39prodindf 30687 . . . . 5 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) = if(ran 𝑐𝐴, 1, 0))
4125, 40eqtr4d 2817 . . . 4 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
4241sumeq2dv 14845 . . 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 2814 . 2 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
44 hashcl 13466 . . . . 5 (((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
459, 44syl 17 . . . 4 (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
4645nn0cnd 11708 . . 3 (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
4746mulid1d 10396 . 2 (𝜑 → ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1) = (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)))
4812, 43, 473eqtr3rd 2823 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 198  wa 386  wo 836   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  cin 3791  wss 3792  ifcif 4307  ran crn 5358  cfv 6137  (class class class)co 6924  Fincfn 8243  cc 10272  0cc0 10274  1c1 10275   · cmul 10279  cn 11378  0cn0 11646  cz 11732  cuz 11996  ..^cfzo 12788  chash 13439  Σcsu 14828  cprod 15042  𝟭cind 30674  reprcrepr 31292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-oi 8706  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11035  df-nn 11379  df-2 11442  df-3 11443  df-n0 11647  df-z 11733  df-uz 11997  df-rp 12142  df-fz 12648  df-fzo 12789  df-seq 13124  df-exp 13183  df-hash 13440  df-cj 14250  df-re 14251  df-im 14252  df-sqrt 14386  df-abs 14387  df-clim 14631  df-sum 14829  df-prod 15043  df-ind 30675  df-repr 31293
This theorem is referenced by:  hashrepr  31309  breprexpnat  31318
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