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Theorem fsumcnv 14346
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 3508 . . . 4 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
2 fvex 6113 . . . . 5 (2nd𝑦) ∈ V
3 fvex 6113 . . . . 5 (1st𝑦) ∈ V
4 opex 4859 . . . . . . 7 𝑗, 𝑘⟩ ∈ V
5 fsumcnv.1 . . . . . . 7 (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
64, 5csbie 3525 . . . . . 6 𝑗, 𝑘⟩ / 𝑥𝐵 = 𝐷
7 opeq12 4342 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd𝑦), (1st𝑦)⟩)
87csbeq1d 3506 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑗, 𝑘⟩ / 𝑥𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
96, 8syl5eqr 2658 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
102, 3, 9csbie2 3529 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵
111, 10syl6eqr 2662 . . 3 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
12 fsumcnv.3 . . . 4 (𝜑𝐴 ∈ Fin)
13 cnvfi 8131 . . . 4 (𝐴 ∈ Fin → 𝐴 ∈ Fin)
1412, 13syl 17 . . 3 (𝜑𝐴 ∈ Fin)
15 relcnv 5422 . . . . 5 Rel 𝐴
16 cnvf1o 7163 . . . . 5 (Rel 𝐴 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
1715, 16ax-mp 5 . . . 4 (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴
18 fsumcnv.4 . . . . . 6 (𝜑 → Rel 𝐴)
19 dfrel2 5502 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
2018, 19sylib 207 . . . . 5 (𝜑𝐴 = 𝐴)
21 f1oeq3 6042 . . . . 5 (𝐴 = 𝐴 → ((𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴 ↔ (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴))
2220, 21syl 17 . . . 4 (𝜑 → ((𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴 ↔ (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴))
2317, 22mpbii 222 . . 3 (𝜑 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
24 1st2nd 7105 . . . . . . 7 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2515, 24mpan 702 . . . . . 6 (𝑦𝐴𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2625fveq2d 6107 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩))
27 id 22 . . . . . . 7 (𝑦𝐴𝑦𝐴)
2825, 27eqeltrrd 2689 . . . . . 6 (𝑦𝐴 → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
29 sneq 4135 . . . . . . . . . 10 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3029cnveqd 5220 . . . . . . . . 9 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3130unieqd 4382 . . . . . . . 8 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
32 opswap 5540 . . . . . . . 8 {⟨(1st𝑦), (2nd𝑦)⟩} = ⟨(2nd𝑦), (1st𝑦)⟩
3331, 32syl6eq 2660 . . . . . . 7 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = ⟨(2nd𝑦), (1st𝑦)⟩)
34 eqid 2610 . . . . . . 7 (𝑧𝐴 {𝑧}) = (𝑧𝐴 {𝑧})
35 opex 4859 . . . . . . 7 ⟨(2nd𝑦), (1st𝑦)⟩ ∈ V
3633, 34, 35fvmpt 6191 . . . . . 6 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
3728, 36syl 17 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
3826, 37eqtrd 2644 . . . 4 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
3938adantl 481 . . 3 ((𝜑𝑦𝐴) → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
40 fsumcnv.5 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
4111, 14, 23, 39, 40fsumf1o 14301 . 2 (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
42 csbeq1a 3508 . . . . 5 (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ → 𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
4325, 42syl 17 . . . 4 (𝑦𝐴𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
44 opex 4859 . . . . . . 7 𝑘, 𝑗⟩ ∈ V
45 fsumcnv.2 . . . . . . 7 (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
4644, 45csbie 3525 . . . . . 6 𝑘, 𝑗⟩ / 𝑦𝐶 = 𝐷
47 opeq12 4342 . . . . . . . 8 ((𝑘 = (1st𝑦) ∧ 𝑗 = (2nd𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
4847ancoms 468 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
4948csbeq1d 3506 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑘, 𝑗⟩ / 𝑦𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
5046, 49syl5eqr 2658 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
512, 3, 50csbie2 3529 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶
5243, 51syl6eqr 2662 . . 3 (𝑦𝐴𝐶 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
5352sumeq2i 14277 . 2 Σ𝑦 𝐴𝐶 = Σ𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷
5441, 53syl6eqr 2662 1 (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  csb 3499  {csn 4125  cop 4131   cuni 4372  cmpt 4643  ccnv 5037  Rel wrel 5043  1-1-ontowf1o 5803  cfv 5804  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  cc 9813  Σcsu 14264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265
This theorem is referenced by:  fsumcom2  14347  fsumcom2OLD  14348
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