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Theorem wemaplem1 9003
Description: Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
wemaplem1 ((𝑃𝑉𝑄𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
Distinct variable groups:   𝑎,𝑏,𝑥   𝑇,𝑎,𝑏   𝑤,𝑎,𝑦,𝑧,𝑏,𝑥,𝐴   𝑃,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑅,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝑆,𝑎,𝑏,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏)

Proof of Theorem wemaplem1
StepHypRef Expression
1 fveq1 6658 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑧) = (𝑃𝑧))
2 fveq1 6658 . . . . . 6 (𝑦 = 𝑄 → (𝑦𝑧) = (𝑄𝑧))
31, 2breqan12d 5069 . . . . 5 ((𝑥 = 𝑃𝑦 = 𝑄) → ((𝑥𝑧)𝑆(𝑦𝑧) ↔ (𝑃𝑧)𝑆(𝑄𝑧)))
4 fveq1 6658 . . . . . . . 8 (𝑥 = 𝑃 → (𝑥𝑤) = (𝑃𝑤))
5 fveq1 6658 . . . . . . . 8 (𝑦 = 𝑄 → (𝑦𝑤) = (𝑄𝑤))
64, 5eqeqan12d 2841 . . . . . . 7 ((𝑥 = 𝑃𝑦 = 𝑄) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑃𝑤) = (𝑄𝑤)))
76imbi2d 344 . . . . . 6 ((𝑥 = 𝑃𝑦 = 𝑄) → ((𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))))
87ralbidv 3192 . . . . 5 ((𝑥 = 𝑃𝑦 = 𝑄) → (∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))))
93, 8anbi12d 633 . . . 4 ((𝑥 = 𝑃𝑦 = 𝑄) → (((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)))))
109rexbidv 3290 . . 3 ((𝑥 = 𝑃𝑦 = 𝑄) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐴 ((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)))))
11 fveq2 6659 . . . . . 6 (𝑧 = 𝑎 → (𝑃𝑧) = (𝑃𝑎))
12 fveq2 6659 . . . . . 6 (𝑧 = 𝑎 → (𝑄𝑧) = (𝑄𝑎))
1311, 12breq12d 5066 . . . . 5 (𝑧 = 𝑎 → ((𝑃𝑧)𝑆(𝑄𝑧) ↔ (𝑃𝑎)𝑆(𝑄𝑎)))
14 breq2 5057 . . . . . . . 8 (𝑧 = 𝑎 → (𝑤𝑅𝑧𝑤𝑅𝑎))
1514imbi1d 345 . . . . . . 7 (𝑧 = 𝑎 → ((𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)) ↔ (𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤))))
1615ralbidv 3192 . . . . . 6 (𝑧 = 𝑎 → (∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)) ↔ ∀𝑤𝐴 (𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤))))
17 breq1 5056 . . . . . . . 8 (𝑤 = 𝑏 → (𝑤𝑅𝑎𝑏𝑅𝑎))
18 fveq2 6659 . . . . . . . . 9 (𝑤 = 𝑏 → (𝑃𝑤) = (𝑃𝑏))
19 fveq2 6659 . . . . . . . . 9 (𝑤 = 𝑏 → (𝑄𝑤) = (𝑄𝑏))
2018, 19eqeq12d 2840 . . . . . . . 8 (𝑤 = 𝑏 → ((𝑃𝑤) = (𝑄𝑤) ↔ (𝑃𝑏) = (𝑄𝑏)))
2117, 20imbi12d 348 . . . . . . 7 (𝑤 = 𝑏 → ((𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤)) ↔ (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏))))
2221cbvralvw 3435 . . . . . 6 (∀𝑤𝐴 (𝑤𝑅𝑎 → (𝑃𝑤) = (𝑄𝑤)) ↔ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))
2316, 22syl6bb 290 . . . . 5 (𝑧 = 𝑎 → (∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤)) ↔ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏))))
2413, 23anbi12d 633 . . . 4 (𝑧 = 𝑎 → (((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))) ↔ ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
2524cbvrexvw 3436 . . 3 (∃𝑧𝐴 ((𝑃𝑧)𝑆(𝑄𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑃𝑤) = (𝑄𝑤))) ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏))))
2610, 25syl6bb 290 . 2 ((𝑥 = 𝑃𝑦 = 𝑄) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
27 wemapso.t . 2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
2826, 27brabga 5409 1 ((𝑃𝑉𝑄𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wrex 3134   class class class wbr 5053  {copab 5115  cfv 6344
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-pr 5318
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-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-iota 6303  df-fv 6352
This theorem is referenced by:  wemaplem2  9004  wemaplem3  9005  wemappo  9006  wemapsolem  9007
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