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Theorem gsummpt2co 28917
Description: Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.)
Hypotheses
Ref Expression
gsummpt2co.b 𝐵 = (Base‘𝑊)
gsummpt2co.z 0 = (0g𝑊)
gsummpt2co.w (𝜑𝑊 ∈ CMnd)
gsummpt2co.a (𝜑𝐴 ∈ Fin)
gsummpt2co.e (𝜑𝐸𝑉)
gsummpt2co.1 ((𝜑𝑥𝐴) → 𝐶𝐵)
gsummpt2co.2 ((𝜑𝑥𝐴) → 𝐷𝐸)
gsummpt2co.3 𝐹 = (𝑥𝐴𝐷)
Assertion
Ref Expression
gsummpt2co (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶)))))
Distinct variable groups:   𝑥, 0 ,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐸,𝑦   𝑥,𝐹,𝑦   𝑦,𝑉   𝑥,𝑊,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑥)   𝐷(𝑥,𝑦)   𝑉(𝑥)

Proof of Theorem gsummpt2co
Dummy variables 𝑧 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 3514 . . . 4 𝑥(2nd𝑝) / 𝑥𝐶
2 gsummpt2co.b . . . 4 𝐵 = (Base‘𝑊)
3 gsummpt2co.z . . . 4 0 = (0g𝑊)
4 csbeq1a 3507 . . . 4 (𝑥 = (2nd𝑝) → 𝐶 = (2nd𝑝) / 𝑥𝐶)
5 gsummpt2co.w . . . 4 (𝜑𝑊 ∈ CMnd)
6 gsummpt2co.a . . . 4 (𝜑𝐴 ∈ Fin)
7 ssid 3586 . . . . 5 𝐵𝐵
87a1i 11 . . . 4 (𝜑𝐵𝐵)
9 gsummpt2co.1 . . . 4 ((𝜑𝑥𝐴) → 𝐶𝐵)
10 elcnv 5209 . . . . . 6 (𝑝𝐹 ↔ ∃𝑧𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧))
11 vex 3175 . . . . . . . . . 10 𝑧 ∈ V
12 vex 3175 . . . . . . . . . 10 𝑥 ∈ V
1311, 12op2ndd 7047 . . . . . . . . 9 (𝑝 = ⟨𝑧, 𝑥⟩ → (2nd𝑝) = 𝑥)
1413adantr 479 . . . . . . . 8 ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd𝑝) = 𝑥)
15 gsummpt2co.3 . . . . . . . . . . 11 𝐹 = (𝑥𝐴𝐷)
1615dmmptss 5534 . . . . . . . . . 10 dom 𝐹𝐴
1712, 11breldm 5238 . . . . . . . . . 10 (𝑥𝐹𝑧𝑥 ∈ dom 𝐹)
1816, 17sseldi 3565 . . . . . . . . 9 (𝑥𝐹𝑧𝑥𝐴)
1918adantl 480 . . . . . . . 8 ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → 𝑥𝐴)
2014, 19eqeltrd 2687 . . . . . . 7 ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd𝑝) ∈ 𝐴)
2120exlimivv 1846 . . . . . 6 (∃𝑧𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd𝑝) ∈ 𝐴)
2210, 21sylbi 205 . . . . 5 (𝑝𝐹 → (2nd𝑝) ∈ 𝐴)
2322adantl 480 . . . 4 ((𝜑𝑝𝐹) → (2nd𝑝) ∈ 𝐴)
2415funmpt2 5827 . . . . . . 7 Fun 𝐹
25 funcnvcnv 5856 . . . . . . 7 (Fun 𝐹 → Fun 𝐹)
2624, 25ax-mp 5 . . . . . 6 Fun 𝐹
2726a1i 11 . . . . 5 ((𝜑𝑥𝐴) → Fun 𝐹)
28 dfdm4 5225 . . . . . . . 8 dom 𝐹 = ran 𝐹
2915dmeqi 5234 . . . . . . . . 9 dom 𝐹 = dom (𝑥𝐴𝐷)
30 gsummpt2co.2 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐷𝐸)
3130ralrimiva 2948 . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴 𝐷𝐸)
32 dmmptg 5535 . . . . . . . . . 10 (∀𝑥𝐴 𝐷𝐸 → dom (𝑥𝐴𝐷) = 𝐴)
3331, 32syl 17 . . . . . . . . 9 (𝜑 → dom (𝑥𝐴𝐷) = 𝐴)
3429, 33syl5eq 2655 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
3528, 34syl5eqr 2657 . . . . . . 7 (𝜑 → ran 𝐹 = 𝐴)
3635eleq2d 2672 . . . . . 6 (𝜑 → (𝑥 ∈ ran 𝐹𝑥𝐴))
3736biimpar 500 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ ran 𝐹)
38 relcnv 5409 . . . . . 6 Rel 𝐹
39 fcnvgreu 28661 . . . . . 6 (((Rel 𝐹 ∧ Fun 𝐹) ∧ 𝑥 ∈ ran 𝐹) → ∃!𝑝 𝐹𝑥 = (2nd𝑝))
4038, 39mpanl1 711 . . . . 5 ((Fun 𝐹𝑥 ∈ ran 𝐹) → ∃!𝑝 𝐹𝑥 = (2nd𝑝))
4127, 37, 40syl2anc 690 . . . 4 ((𝜑𝑥𝐴) → ∃!𝑝 𝐹𝑥 = (2nd𝑝))
421, 2, 3, 4, 5, 6, 8, 9, 23, 41gsummptf1o 18131 . . 3 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑝𝐹(2nd𝑝) / 𝑥𝐶)))
4315rnmptss 6284 . . . . . . . 8 (∀𝑥𝐴 𝐷𝐸 → ran 𝐹𝐸)
4431, 43syl 17 . . . . . . 7 (𝜑 → ran 𝐹𝐸)
45 dfcnv2 28665 . . . . . . 7 (ran 𝐹𝐸𝐹 = 𝑧𝐸 ({𝑧} × (𝐹 “ {𝑧})))
4644, 45syl 17 . . . . . 6 (𝜑𝐹 = 𝑧𝐸 ({𝑧} × (𝐹 “ {𝑧})))
4746mpteq1d 4660 . . . . 5 (𝜑 → (𝑝𝐹(2nd𝑝) / 𝑥𝐶) = (𝑝 𝑧𝐸 ({𝑧} × (𝐹 “ {𝑧})) ↦ (2nd𝑝) / 𝑥𝐶))
48 nfcv 2750 . . . . . 6 𝑧(2nd𝑝) / 𝑥𝐶
49 csbeq1 3501 . . . . . . . 8 ((2nd𝑝) = 𝑥(2nd𝑝) / 𝑥𝐶 = 𝑥 / 𝑥𝐶)
5013, 49syl 17 . . . . . . 7 (𝑝 = ⟨𝑧, 𝑥⟩ → (2nd𝑝) / 𝑥𝐶 = 𝑥 / 𝑥𝐶)
51 csbid 3506 . . . . . . 7 𝑥 / 𝑥𝐶 = 𝐶
5250, 51syl6eq 2659 . . . . . 6 (𝑝 = ⟨𝑧, 𝑥⟩ → (2nd𝑝) / 𝑥𝐶 = 𝐶)
5348, 1, 52mpt2mptxf 28666 . . . . 5 (𝑝 𝑧𝐸 ({𝑧} × (𝐹 “ {𝑧})) ↦ (2nd𝑝) / 𝑥𝐶) = (𝑧𝐸, 𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶)
5447, 53syl6eq 2659 . . . 4 (𝜑 → (𝑝𝐹(2nd𝑝) / 𝑥𝐶) = (𝑧𝐸, 𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶))
5554oveq2d 6543 . . 3 (𝜑 → (𝑊 Σg (𝑝𝐹(2nd𝑝) / 𝑥𝐶)) = (𝑊 Σg (𝑧𝐸, 𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶)))
56 gsummpt2co.e . . . 4 (𝜑𝐸𝑉)
57 mptfi 8125 . . . . . . . 8 (𝐴 ∈ Fin → (𝑥𝐴𝐷) ∈ Fin)
5815, 57syl5eqel 2691 . . . . . . 7 (𝐴 ∈ Fin → 𝐹 ∈ Fin)
59 cnvfi 8108 . . . . . . 7 (𝐹 ∈ Fin → 𝐹 ∈ Fin)
606, 58, 593syl 18 . . . . . 6 (𝜑𝐹 ∈ Fin)
61 imaexg 6972 . . . . . 6 (𝐹 ∈ Fin → (𝐹 “ {𝑧}) ∈ V)
6260, 61syl 17 . . . . 5 (𝜑 → (𝐹 “ {𝑧}) ∈ V)
6362adantr 479 . . . 4 ((𝜑𝑧𝐸) → (𝐹 “ {𝑧}) ∈ V)
64 simpll 785 . . . . . 6 (((𝜑𝑧𝐸) ∧ 𝑥 ∈ (𝐹 “ {𝑧})) → 𝜑)
65 imassrn 5383 . . . . . . . . 9 (𝐹 “ {𝑧}) ⊆ ran 𝐹
6665, 28sseqtr4i 3600 . . . . . . . 8 (𝐹 “ {𝑧}) ⊆ dom 𝐹
6766, 16sstri 3576 . . . . . . 7 (𝐹 “ {𝑧}) ⊆ 𝐴
6811, 12elimasn 5396 . . . . . . . . . 10 (𝑥 ∈ (𝐹 “ {𝑧}) ↔ ⟨𝑧, 𝑥⟩ ∈ 𝐹)
6968biimpi 204 . . . . . . . . 9 (𝑥 ∈ (𝐹 “ {𝑧}) → ⟨𝑧, 𝑥⟩ ∈ 𝐹)
7069adantl 480 . . . . . . . 8 (((𝜑𝑧𝐸) ∧ 𝑥 ∈ (𝐹 “ {𝑧})) → ⟨𝑧, 𝑥⟩ ∈ 𝐹)
7170, 68sylibr 222 . . . . . . 7 (((𝜑𝑧𝐸) ∧ 𝑥 ∈ (𝐹 “ {𝑧})) → 𝑥 ∈ (𝐹 “ {𝑧}))
7267, 71sseldi 3565 . . . . . 6 (((𝜑𝑧𝐸) ∧ 𝑥 ∈ (𝐹 “ {𝑧})) → 𝑥𝐴)
7364, 72, 9syl2anc 690 . . . . 5 (((𝜑𝑧𝐸) ∧ 𝑥 ∈ (𝐹 “ {𝑧})) → 𝐶𝐵)
7473anasss 676 . . . 4 ((𝜑 ∧ (𝑧𝐸𝑥 ∈ (𝐹 “ {𝑧}))) → 𝐶𝐵)
75 df-br 4578 . . . . . . . . 9 (𝑧𝐹𝑥 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝐹)
7670, 75sylibr 222 . . . . . . . 8 (((𝜑𝑧𝐸) ∧ 𝑥 ∈ (𝐹 “ {𝑧})) → 𝑧𝐹𝑥)
7776anasss 676 . . . . . . 7 ((𝜑 ∧ (𝑧𝐸𝑥 ∈ (𝐹 “ {𝑧}))) → 𝑧𝐹𝑥)
7877pm2.24d 145 . . . . . 6 ((𝜑 ∧ (𝑧𝐸𝑥 ∈ (𝐹 “ {𝑧}))) → (¬ 𝑧𝐹𝑥𝐶 = 0 ))
7978imp 443 . . . . 5 (((𝜑 ∧ (𝑧𝐸𝑥 ∈ (𝐹 “ {𝑧}))) ∧ ¬ 𝑧𝐹𝑥) → 𝐶 = 0 )
8079anasss 676 . . . 4 ((𝜑 ∧ ((𝑧𝐸𝑥 ∈ (𝐹 “ {𝑧})) ∧ ¬ 𝑧𝐹𝑥)) → 𝐶 = 0 )
812, 3, 5, 56, 63, 74, 60, 80gsum2d2 18142 . . 3 (𝜑 → (𝑊 Σg (𝑧𝐸, 𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶)) = (𝑊 Σg (𝑧𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶)))))
8242, 55, 813eqtrd 2647 . 2 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑧𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶)))))
83 nfcv 2750 . . . 4 𝑧(𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶))
84 nfcv 2750 . . . 4 𝑦(𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶))
85 sneq 4134 . . . . . . 7 (𝑦 = 𝑧 → {𝑦} = {𝑧})
8685imaeq2d 5372 . . . . . 6 (𝑦 = 𝑧 → (𝐹 “ {𝑦}) = (𝐹 “ {𝑧}))
8786mpteq1d 4660 . . . . 5 (𝑦 = 𝑧 → (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶) = (𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶))
8887oveq2d 6543 . . . 4 (𝑦 = 𝑧 → (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶)))
8983, 84, 88cbvmpt 4671 . . 3 (𝑦𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶))) = (𝑧𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶)))
9089oveq2i 6538 . 2 (𝑊 Σg (𝑦𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶)))) = (𝑊 Σg (𝑧𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑧}) ↦ 𝐶))))
9182, 90syl6eqr 2661 1 (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1976  wral 2895  ∃!wreu 2897  Vcvv 3172  csb 3498  wss 3539  {csn 4124  cop 4130   ciun 4449   class class class wbr 4577  cmpt 4637   × cxp 5026  ccnv 5027  dom cdm 5028  ran crn 5029  cima 5031  Rel wrel 5033  Fun wfun 5784  cfv 5790  (class class class)co 6527  cmpt2 6529  2nd c2nd 7035  Fincfn 7818  Basecbs 15641  0gc0g 15869   Σg cgsu 15870  CMndccmn 17962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-seq 12619  df-hash 12935  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-0g 15871  df-gsum 15872  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-submnd 17105  df-mulg 17310  df-cntz 17519  df-cmn 17964
This theorem is referenced by:  gsummpt2d  28918
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