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Theorem fsumcnv 11213
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 3012 . . . 4 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
2 2ndexg 6066 . . . . . 6 (𝑦 ∈ V → (2nd𝑦) ∈ V)
32elv 2690 . . . . 5 (2nd𝑦) ∈ V
4 1stexg 6065 . . . . . 6 (𝑦 ∈ V → (1st𝑦) ∈ V)
54elv 2690 . . . . 5 (1st𝑦) ∈ V
6 vex 2689 . . . . . . . 8 𝑗 ∈ V
7 vex 2689 . . . . . . . 8 𝑘 ∈ V
86, 7opex 4151 . . . . . . 7 𝑗, 𝑘⟩ ∈ V
9 fsumcnv.1 . . . . . . 7 (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
108, 9csbie 3045 . . . . . 6 𝑗, 𝑘⟩ / 𝑥𝐵 = 𝐷
11 opeq12 3707 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd𝑦), (1st𝑦)⟩)
1211csbeq1d 3010 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑗, 𝑘⟩ / 𝑥𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
1310, 12syl5eqr 2186 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
143, 5, 13csbie2 3049 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵
151, 14syl6eqr 2190 . . 3 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
16 fsumcnv.4 . . . 4 (𝜑 → Rel 𝐴)
17 fsumcnv.3 . . . 4 (𝜑𝐴 ∈ Fin)
18 relcnvfi 6829 . . . 4 ((Rel 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)
1916, 17, 18syl2anc 408 . . 3 (𝜑𝐴 ∈ Fin)
20 relcnv 4917 . . . . 5 Rel 𝐴
21 cnvf1o 6122 . . . . 5 (Rel 𝐴 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
2220, 21ax-mp 5 . . . 4 (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴
23 dfrel2 4989 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
2416, 23sylib 121 . . . . 5 (𝜑𝐴 = 𝐴)
25 f1oeq3 5358 . . . . 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 147 . . 3 (𝜑 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
28 1st2nd 6079 . . . . . . 7 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2920, 28mpan 420 . . . . . 6 (𝑦𝐴𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3029fveq2d 5425 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩))
31 id 19 . . . . . . 7 (𝑦𝐴𝑦𝐴)
3229, 31eqeltrrd 2217 . . . . . 6 (𝑦𝐴 → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
33 sneq 3538 . . . . . . . . . 10 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3433cnveqd 4715 . . . . . . . . 9 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3534unieqd 3747 . . . . . . . 8 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
36 opswapg 5025 . . . . . . . . 9 (((1st𝑦) ∈ V ∧ (2nd𝑦) ∈ V) → {⟨(1st𝑦), (2nd𝑦)⟩} = ⟨(2nd𝑦), (1st𝑦)⟩)
375, 3, 36mp2an 422 . . . . . . . 8 {⟨(1st𝑦), (2nd𝑦)⟩} = ⟨(2nd𝑦), (1st𝑦)⟩
3835, 37syl6eq 2188 . . . . . . 7 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = ⟨(2nd𝑦), (1st𝑦)⟩)
39 eqid 2139 . . . . . . 7 (𝑧𝐴 {𝑧}) = (𝑧𝐴 {𝑧})
403, 5opex 4151 . . . . . . 7 ⟨(2nd𝑦), (1st𝑦)⟩ ∈ V
4138, 39, 40fvmpt 5498 . . . . . 6 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
4232, 41syl 14 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
4330, 42eqtrd 2172 . . . 4 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
4443adantl 275 . . 3 ((𝜑𝑦𝐴) → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
45 fsumcnv.5 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
4615, 19, 27, 44, 45fsumf1o 11166 . 2 (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
47 csbeq1a 3012 . . . . 5 (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ → 𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
4829, 47syl 14 . . . 4 (𝑦𝐴𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
497, 6opex 4151 . . . . . . 7 𝑘, 𝑗⟩ ∈ V
50 fsumcnv.2 . . . . . . 7 (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
5149, 50csbie 3045 . . . . . 6 𝑘, 𝑗⟩ / 𝑦𝐶 = 𝐷
52 opeq12 3707 . . . . . . . 8 ((𝑘 = (1st𝑦) ∧ 𝑗 = (2nd𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5352ancoms 266 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5453csbeq1d 3010 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑘, 𝑗⟩ / 𝑦𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
5551, 54syl5eqr 2186 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
563, 5, 55csbie2 3049 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶
5748, 56syl6eqr 2190 . . 3 (𝑦𝐴𝐶 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
5857sumeq2i 11140 . 2 Σ𝑦 𝐴𝐶 = Σ𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷
5946, 58syl6eqr 2190 1 (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  Vcvv 2686  csb 3003  {csn 3527  cop 3530   cuni 3736  cmpt 3989  ccnv 4538  Rel wrel 4544  1-1-ontowf1o 5122  cfv 5123  1st c1st 6036  2nd c2nd 6037  Fincfn 6634  cc 7625  Σcsu 11129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-ihash 10529  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130
This theorem is referenced by:  fisumcom2  11214
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