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Theorem hashreprin 31038
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 31034 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) ∈ Fin)
6 inss2 3982 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
76a1i 11 . . . . 5 (𝜑 → (𝐴𝐵) ⊆ 𝐵)
81, 2, 3, 7reprss 31035 . . . 4 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) ⊆ (𝐵(repr‘𝑆)𝑀))
95, 8ssfid 8343 . . 3 (𝜑 → ((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin)
10 1cnd 10262 . . 3 (𝜑 → 1 ∈ ℂ)
11 fsumconst 14729 . . 3 ((((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1))
129, 10, 11syl2anc 573 . 2 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1))
1310ralrimivw 3116 . . . 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 14665 . . . 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 1475 . . 3 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0))
171, 2, 3reprinrn 31036 . . . . . . . 8 (𝜑 → (𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)))
18 incom 3956 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
1918oveq1i 6806 . . . . . . . . . . 11 ((𝐵𝐴)(repr‘𝑆)𝑀) = ((𝐴𝐵)(repr‘𝑆)𝑀)
2019eleq2i 2842 . . . . . . . . . 10 (𝑐 ∈ ((𝐵𝐴)(repr‘𝑆)𝑀) ↔ 𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀))
2120bibi1i 327 . . . . . . . . 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 325 . . . . . . . 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 220 . . . . . . 7 (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐵(repr‘𝑆)𝑀) ∧ ran 𝑐𝐴)))
2423baibd 529 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ ran 𝑐𝐴))
2524ifbid 4248 . . . . 5 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0) = if(ran 𝑐𝐴, 1, 0))
26 nnex 11232 . . . . . . . . 9 ℕ ∈ V
2726a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
2827ralrimivw 3116 . . . . . . 7 (𝜑 → ∀𝑐 ∈ (𝐵(repr‘𝑆)𝑀)ℕ ∈ V)
2928r19.21bi 3081 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ℕ ∈ V)
30 fzofi 12981 . . . . . . 7 (0..^𝑆) ∈ Fin
3130a1i 11 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → (0..^𝑆) ∈ Fin)
32 reprval.a . . . . . . 7 (𝜑𝐴 ⊆ ℕ)
3332adantr 466 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
341adantr 466 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝐵 ⊆ ℕ)
352adantr 466 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
363adantr 466 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
37 simpr 471 . . . . . . . 8 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐 ∈ (𝐵(repr‘𝑆)𝑀))
3834, 35, 36, 37reprf 31030 . . . . . . 7 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶𝐵)
3938, 34fssd 6198 . . . . . 6 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → 𝑐:(0..^𝑆)⟶ℕ)
4029, 31, 33, 39prodindf 30425 . . . . 5 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)) = if(ran 𝑐𝐴, 1, 0))
4125, 40eqtr4d 2808 . . . 4 ((𝜑𝑐 ∈ (𝐵(repr‘𝑆)𝑀)) → if(𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀), 1, 0) = ∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
4241sumeq2dv 14641 . . 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 2805 . 2 (𝜑 → Σ𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀)1 = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))
44 hashcl 13349 . . . . 5 (((𝐴𝐵)(repr‘𝑆)𝑀) ∈ Fin → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
459, 44syl 17 . . . 4 (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℕ0)
4645nn0cnd 11560 . . 3 (𝜑 → (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) ∈ ℂ)
4746mulid1d 10263 . 2 (𝜑 → ((♯‘((𝐴𝐵)(repr‘𝑆)𝑀)) · 1) = (♯‘((𝐴𝐵)(repr‘𝑆)𝑀)))
4812, 43, 473eqtr3rd 2814 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 196  wa 382  wo 836   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cin 3722  wss 3723  ifcif 4226  ran crn 5251  cfv 6030  (class class class)co 6796  Fincfn 8113  cc 10140  0cc0 10142  1c1 10143   · cmul 10147  cn 11226  0cn0 11499  cz 11584  cuz 11893  ..^cfzo 12673  chash 13321  Σcsu 14624  cprod 14842  𝟭cind 30412  reprcrepr 31026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-oi 8575  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-rp 12036  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-prod 14843  df-ind 30413  df-repr 31027
This theorem is referenced by:  hashrepr  31043  breprexpnat  31052
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