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Theorem fprodcnv 16016
Description: Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
Hypotheses
Ref Expression
fprodcnv.1 (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
fprodcnv.2 (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
fprodcnv.3 (𝜑𝐴 ∈ Fin)
fprodcnv.4 (𝜑 → Rel 𝐴)
fprodcnv.5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
fprodcnv (𝜑 → ∏𝑥𝐴 𝐵 = ∏𝑦 𝐴𝐶)
Distinct variable groups:   𝑥,𝐴,𝑦   𝐵,𝑗,𝑘,𝑦   𝐶,𝑗,𝑘   𝑥,𝐷,𝑦   𝑗,𝑘,𝑥,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑗,𝑘)   𝐴(𝑗,𝑘)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑗,𝑘)

Proof of Theorem fprodcnv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbeq1a 3922 . . . 4 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
2 fvex 6920 . . . . 5 (2nd𝑦) ∈ V
3 fvex 6920 . . . . 5 (1st𝑦) ∈ V
4 opex 5475 . . . . . . 7 𝑗, 𝑘⟩ ∈ V
5 fprodcnv.1 . . . . . . 7 (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
64, 5csbie 3944 . . . . . 6 𝑗, 𝑘⟩ / 𝑥𝐵 = 𝐷
7 opeq12 4880 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd𝑦), (1st𝑦)⟩)
87csbeq1d 3912 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑗, 𝑘⟩ / 𝑥𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
96, 8eqtr3id 2789 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
102, 3, 9csbie2 3950 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵
111, 10eqtr4di 2793 . . 3 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
12 fprodcnv.3 . . . 4 (𝜑𝐴 ∈ Fin)
13 cnvfi 9215 . . . 4 (𝐴 ∈ Fin → 𝐴 ∈ Fin)
1412, 13syl 17 . . 3 (𝜑𝐴 ∈ Fin)
15 relcnv 6125 . . . . 5 Rel 𝐴
16 cnvf1o 8135 . . . . 5 (Rel 𝐴 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
1715, 16ax-mp 5 . . . 4 (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴
18 fprodcnv.4 . . . . . 6 (𝜑 → Rel 𝐴)
19 dfrel2 6211 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
2018, 19sylib 218 . . . . 5 (𝜑𝐴 = 𝐴)
2120f1oeq3d 6846 . . . 4 (𝜑 → ((𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴 ↔ (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴))
2217, 21mpbii 233 . . 3 (𝜑 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
23 1st2nd 8063 . . . . . . 7 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2415, 23mpan 690 . . . . . 6 (𝑦𝐴𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2524fveq2d 6911 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩))
2624eleq1d 2824 . . . . . . 7 (𝑦𝐴 → (𝑦𝐴 ↔ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴))
2726ibi 267 . . . . . 6 (𝑦𝐴 → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
28 sneq 4641 . . . . . . . . . 10 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
2928cnveqd 5889 . . . . . . . . 9 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3029unieqd 4925 . . . . . . . 8 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
31 opswap 6251 . . . . . . . 8 {⟨(1st𝑦), (2nd𝑦)⟩} = ⟨(2nd𝑦), (1st𝑦)⟩
3230, 31eqtrdi 2791 . . . . . . 7 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = ⟨(2nd𝑦), (1st𝑦)⟩)
33 eqid 2735 . . . . . . 7 (𝑧𝐴 {𝑧}) = (𝑧𝐴 {𝑧})
34 opex 5475 . . . . . . 7 ⟨(2nd𝑦), (1st𝑦)⟩ ∈ V
3532, 33, 34fvmpt 7016 . . . . . 6 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
3627, 35syl 17 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
3725, 36eqtrd 2775 . . . 4 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
3837adantl 481 . . 3 ((𝜑𝑦𝐴) → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
39 fprodcnv.5 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
4011, 14, 22, 38, 39fprodf1o 15979 . 2 (𝜑 → ∏𝑥𝐴 𝐵 = ∏𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
41 csbeq1a 3922 . . . . 5 (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ → 𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
4224, 41syl 17 . . . 4 (𝑦𝐴𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
43 opex 5475 . . . . . . 7 𝑘, 𝑗⟩ ∈ V
44 fprodcnv.2 . . . . . . 7 (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
4543, 44csbie 3944 . . . . . 6 𝑘, 𝑗⟩ / 𝑦𝐶 = 𝐷
46 opeq12 4880 . . . . . . . 8 ((𝑘 = (1st𝑦) ∧ 𝑗 = (2nd𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
4746ancoms 458 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
4847csbeq1d 3912 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑘, 𝑗⟩ / 𝑦𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
4945, 48eqtr3id 2789 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
502, 3, 49csbie2 3950 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶
5142, 50eqtr4di 2793 . . 3 (𝑦𝐴𝐶 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
5251prodeq2i 15951 . 2 𝑦 𝐴𝐶 = ∏𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷
5340, 52eqtr4di 2793 1 (𝜑 → ∏𝑥𝐴 𝐵 = ∏𝑦 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  csb 3908  {csn 4631  cop 4637   cuni 4912  cmpt 5231  ccnv 5688  Rel wrel 5694  1-1-ontowf1o 6562  cfv 6563  1st c1st 8011  2nd c2nd 8012  Fincfn 8984  cc 11151  cprod 15936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-prod 15937
This theorem is referenced by:  fprodcom2  16017
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