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

Theorem mposn 7795
 Description: An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
mposn.f 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
mposn.a (𝑥 = 𝐴𝐶 = 𝐷)
mposn.b (𝑦 = 𝐵𝐷 = 𝐸)
Assertion
Ref Expression
mposn ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐸,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mposn
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 xpsng 6893 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
213adant3 1129 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
32mpteq1d 5142 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶) = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
4 mposn.f . . . 4 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
5 mpompts 7759 . . . 4 (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
64, 5eqtri 2847 . . 3 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
76a1i 11 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
8 op2ndg 7698 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 fveq2 6662 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = (2nd ‘⟨𝐴, 𝐵⟩))
109eqcomd 2830 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘⟨𝐴, 𝐵⟩) = (2nd𝑝))
1110eqeq1d 2826 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐵 ↔ (2nd𝑝) = 𝐵))
128, 11syl5ibcom 248 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
13123adant3 1129 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
1413imp 410 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (2nd𝑝) = 𝐵)
15 op1stg 7697 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
16 fveq2 6662 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = (1st ‘⟨𝐴, 𝐵⟩))
1716eqcomd 2830 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘⟨𝐴, 𝐵⟩) = (1st𝑝))
1817eqeq1d 2826 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐴 ↔ (1st𝑝) = 𝐴))
1915, 18syl5ibcom 248 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
20193adant3 1129 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
2120imp 410 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) = 𝐴)
22 simp1 1133 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴𝑉)
23 simpl2 1189 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵𝑊)
24 mposn.a . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
2524adantl 485 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
26 mposn.b . . . . . . . . 9 (𝑦 = 𝐵𝐷 = 𝐸)
2725, 26sylan9eq 2879 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐸)
2823, 27csbied 3903 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐸)
2922, 28csbied 3903 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
3029adantr 484 . . . . 5 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
31 csbeq1 3870 . . . . . . . 8 ((1st𝑝) = 𝐴(1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶)
3231eqeq1d 2826 . . . . . . 7 ((1st𝑝) = 𝐴 → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
3332adantl 485 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
34 csbeq1 3870 . . . . . . . . 9 ((2nd𝑝) = 𝐵(2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3534adantr 484 . . . . . . . 8 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3635csbeq2dv 3874 . . . . . . 7 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
3736eqeq1d 2826 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸))
3833, 37bitrd 282 . . . . 5 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸))
3930, 38syl5ibrcom 250 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
4014, 21, 39mp2and 698 . . 3 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸)
41 opex 5344 . . . 4 𝐴, 𝐵⟩ ∈ V
4241a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ⟨𝐴, 𝐵⟩ ∈ V)
43 simp3 1135 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐸𝑈)
4440, 42, 43fmptsnd 6923 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → {⟨⟨𝐴, 𝐵⟩, 𝐸⟩} = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
453, 7, 443eqtr4d 2869 1 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  Vcvv 3481  ⦋csb 3867  {csn 4551  ⟨cop 4557   ↦ cmpt 5133   × cxp 5541  ‘cfv 6344   ∈ cmpo 7152  1st c1st 7683  2nd c2nd 7684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686 This theorem is referenced by:  mat1dim0  21085  mat1dimid  21086  mat1dimmul  21088  d1mat2pmat  21350
 Copyright terms: Public domain W3C validator