MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fprodcnv Structured version   Visualization version   GIF version

Theorem fprodcnv 14996
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 3700 . . . 4 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
2 fvex 6388 . . . . 5 (2nd𝑦) ∈ V
3 fvex 6388 . . . . 5 (1st𝑦) ∈ V
4 opex 5088 . . . . . . 7 𝑗, 𝑘⟩ ∈ V
5 fprodcnv.1 . . . . . . 7 (𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)
64, 5csbie 3717 . . . . . 6 𝑗, 𝑘⟩ / 𝑥𝐵 = 𝐷
7 opeq12 4561 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑗, 𝑘⟩ = ⟨(2nd𝑦), (1st𝑦)⟩)
87csbeq1d 3698 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑗, 𝑘⟩ / 𝑥𝐵 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
96, 8syl5eqr 2813 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵)
102, 3, 9csbie2 3721 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(2nd𝑦), (1st𝑦)⟩ / 𝑥𝐵
111, 10syl6eqr 2817 . . 3 (𝑥 = ⟨(2nd𝑦), (1st𝑦)⟩ → 𝐵 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
12 fprodcnv.3 . . . 4 (𝜑𝐴 ∈ Fin)
13 cnvfi 8455 . . . 4 (𝐴 ∈ Fin → 𝐴 ∈ Fin)
1412, 13syl 17 . . 3 (𝜑𝐴 ∈ Fin)
15 relcnv 5685 . . . . 5 Rel 𝐴
16 cnvf1o 7478 . . . . 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 5766 . . . . . 6 (Rel 𝐴𝐴 = 𝐴)
2018, 19sylib 209 . . . . 5 (𝜑𝐴 = 𝐴)
21 f1oeq3 6312 . . . . 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 224 . . 3 (𝜑 → (𝑧𝐴 {𝑧}):𝐴1-1-onto𝐴)
24 1st2nd 7414 . . . . . . 7 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2515, 24mpan 681 . . . . . 6 (𝑦𝐴𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2625fveq2d 6379 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩))
2725eleq1d 2829 . . . . . . 7 (𝑦𝐴 → (𝑦𝐴 ↔ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴))
2827ibi 258 . . . . . 6 (𝑦𝐴 → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
29 sneq 4344 . . . . . . . . . 10 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3029cnveqd 5466 . . . . . . . . 9 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
3130unieqd 4604 . . . . . . . 8 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = {⟨(1st𝑦), (2nd𝑦)⟩})
32 opswap 5808 . . . . . . . 8 {⟨(1st𝑦), (2nd𝑦)⟩} = ⟨(2nd𝑦), (1st𝑦)⟩
3331, 32syl6eq 2815 . . . . . . 7 (𝑧 = ⟨(1st𝑦), (2nd𝑦)⟩ → {𝑧} = ⟨(2nd𝑦), (1st𝑦)⟩)
34 eqid 2765 . . . . . . 7 (𝑧𝐴 {𝑧}) = (𝑧𝐴 {𝑧})
35 opex 5088 . . . . . . 7 ⟨(2nd𝑦), (1st𝑦)⟩ ∈ V
3633, 34, 35fvmpt 6471 . . . . . 6 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
3728, 36syl 17 . . . . 5 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘⟨(1st𝑦), (2nd𝑦)⟩) = ⟨(2nd𝑦), (1st𝑦)⟩)
3826, 37eqtrd 2799 . . . 4 (𝑦𝐴 → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
3938adantl 473 . . 3 ((𝜑𝑦𝐴) → ((𝑧𝐴 {𝑧})‘𝑦) = ⟨(2nd𝑦), (1st𝑦)⟩)
40 fprodcnv.5 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
4111, 14, 23, 39, 40fprodf1o 14959 . 2 (𝜑 → ∏𝑥𝐴 𝐵 = ∏𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
42 csbeq1a 3700 . . . . 5 (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ → 𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
4325, 42syl 17 . . . 4 (𝑦𝐴𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
44 opex 5088 . . . . . . 7 𝑘, 𝑗⟩ ∈ V
45 fprodcnv.2 . . . . . . 7 (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)
4644, 45csbie 3717 . . . . . 6 𝑘, 𝑗⟩ / 𝑦𝐶 = 𝐷
47 opeq12 4561 . . . . . . . 8 ((𝑘 = (1st𝑦) ∧ 𝑗 = (2nd𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
4847ancoms 450 . . . . . . 7 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → ⟨𝑘, 𝑗⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
4948csbeq1d 3698 . . . . . 6 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝑘, 𝑗⟩ / 𝑦𝐶 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
5046, 49syl5eqr 2813 . . . . 5 ((𝑗 = (2nd𝑦) ∧ 𝑘 = (1st𝑦)) → 𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶)
512, 3, 50csbie2 3721 . . . 4 (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷 = ⟨(1st𝑦), (2nd𝑦)⟩ / 𝑦𝐶
5243, 51syl6eqr 2817 . . 3 (𝑦𝐴𝐶 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷)
5352prodeq2i 14932 . 2 𝑦 𝐴𝐶 = ∏𝑦 𝐴(2nd𝑦) / 𝑗(1st𝑦) / 𝑘𝐷
5441, 53syl6eqr 2817 1 (𝜑 → ∏𝑥𝐴 𝐵 = ∏𝑦 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  csb 3691  {csn 4334  cop 4340   cuni 4594  cmpt 4888  ccnv 5276  Rel wrel 5282  1-1-ontowf1o 6067  cfv 6068  1st c1st 7364  2nd c2nd 7365  Fincfn 8160  cc 10187  cprod 14918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-prod 14919
This theorem is referenced by:  fprodcom2  14997
  Copyright terms: Public domain W3C validator