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Theorem fsumcnv 10831
Description: Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
fsumcnv.1 (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
fsumcnv.2 (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
fsumcnv.3 (𝜑𝐴 ∈ Fin)
fsumcnv.4 (𝜑 → Rel 𝐴)
fsumcnv.5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
fsumcnv (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑗,𝑘,𝑦,𝐵   𝑥,𝑗,𝐶,𝑘   𝜑,𝑥,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑗,𝑘)   𝐴(𝑗,𝑘)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑗,𝑘)

Proof of Theorem fsumcnv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbeq1a 2941 . . . 4 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
2 2ndexg 5939 . . . . . 6 (𝑦 ∈ V → (2nd𝑦) ∈ V)
32elv 2623 . . . . 5 (2nd𝑦) ∈ V
4 1stexg 5938 . . . . . 6 (𝑦 ∈ V → (1st𝑦) ∈ V)
54elv 2623 . . . . 5 (1st𝑦) ∈ V
6 vex 2622 . . . . . . . 8 𝑗 ∈ V
7 vex 2622 . . . . . . . 8 𝑘 ∈ V
86, 7opex 4056 . . . . . . 7 𝑗, 𝑘⟩ ∈ V
9 fsumcnv.1 . . . . . . 7 (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
108, 9csbie 2973 . . . . . 6 𝑗, 𝑘⟩ / 𝑥𝐵 = 𝐷
11 opeq12 3624 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd𝑦), (1st𝑦)⟩)
1211csbeq1d 2939 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑗, 𝑘⟩ / 𝑥𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
1310, 12syl5eqr 2134 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
143, 5, 13csbie2 2977 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵
151, 14syl6eqr 2138 . . 3 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
16 fsumcnv.4 . . . 4 (𝜑 → Rel 𝐴)
17 fsumcnv.3 . . . 4 (𝜑𝐴 ∈ Fin)
18 relcnvfi 6650 . . . 4 ((Rel 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)
1916, 17, 18syl2anc 403 . . 3 (𝜑𝐴 ∈ Fin)
20 relcnv 4810 . . . . 5 Rel 𝐴
21 cnvf1o 5990 . . . . 5 (Rel 𝐴 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
2220, 21ax-mp 7 . . . 4 (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴
23 dfrel2 4881 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
2416, 23sylib 120 . . . . 5 (𝜑𝐴 = 𝐴)
25 f1oeq3 5246 . . . . 5 (𝐴 = 𝐴 → ((𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴 ↔ (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴))
2624, 25syl 14 . . . 4 (𝜑 → ((𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴 ↔ (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴))
2722, 26mpbii 146 . . 3 (𝜑 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
28 1st2nd 5951 . . . . . . 7 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2920, 28mpan 415 . . . . . 6 (𝑦𝐴𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3029fveq2d 5309 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩))
31 id 19 . . . . . . 7 (𝑦𝐴𝑦𝐴)
3229, 31eqeltrrd 2165 . . . . . 6 (𝑦𝐴 → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
33 sneq 3457 . . . . . . . . . 10 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3433cnveqd 4612 . . . . . . . . 9 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3534unieqd 3664 . . . . . . . 8 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
36 opswapg 4917 . . . . . . . . 9 (((1st𝑦) ∈ V ∧ (2nd𝑦) ∈ V) → {⟨(1st𝑦), (2nd𝑦)⟩} = ⟨(2nd𝑦), (1st𝑦)⟩)
375, 3, 36mp2an 417 . . . . . . . 8 {⟨(1st𝑦), (2nd𝑦)⟩} = ⟨(2nd𝑦), (1st𝑦)⟩
3835, 37syl6eq 2136 . . . . . . 7 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = ⟨(2nd𝑦), (1st𝑦)⟩)
39 eqid 2088 . . . . . . 7 (𝑧𝐴 {𝑧}) = (𝑧𝐴 {𝑧})
403, 5opex 4056 . . . . . . 7 ⟨(2nd𝑦), (1st𝑦)⟩ ∈ V
4138, 39, 40fvmpt 5381 . . . . . 6 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
4232, 41syl 14 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
4330, 42eqtrd 2120 . . . 4 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
4443adantl 271 . . 3 ((𝜑𝑦𝐴) → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
45 fsumcnv.5 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
4615, 19, 27, 44, 45fsumf1o 10782 . 2 (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
47 csbeq1a 2941 . . . . 5 (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ → 𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
4829, 47syl 14 . . . 4 (𝑦𝐴𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
497, 6opex 4056 . . . . . . 7 𝑘, 𝑗⟩ ∈ V
50 fsumcnv.2 . . . . . . 7 (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
5149, 50csbie 2973 . . . . . 6 𝑘, 𝑗⟩ / 𝑦𝐶 = 𝐷
52 opeq12 3624 . . . . . . . 8 ((𝑘 = (1st𝑦) ∧ 𝑗 = (2nd𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5352ancoms 264 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5453csbeq1d 2939 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑘, 𝑗⟩ / 𝑦𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
5551, 54syl5eqr 2134 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
563, 5, 55csbie2 2977 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶
5748, 56syl6eqr 2138 . . 3 (𝑦𝐴𝐶 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
5857sumeq2i 10753 . 2 Σ𝑦 𝐴𝐶 = Σ𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷
5946, 58syl6eqr 2138 1 (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619  csb 2933  {csn 3446  cop 3449   cuni 3653  cmpt 3899  ccnv 4437  Rel wrel 4443  1-1-ontowf1o 5014  cfv 5015  1st c1st 5909  2nd c2nd 5910  Fincfn 6457  cc 7348  Σcsu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by:  fisumcom2  10832
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