Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgcoval Structured version   Visualization version   GIF version

Theorem mgcoval 32734
Description: Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
Hypotheses
Ref Expression
mgcoval.1 𝐴 = (Base‘𝑉)
mgcoval.2 𝐵 = (Base‘𝑊)
mgcoval.3 = (le‘𝑉)
mgcoval.4 = (le‘𝑊)
Assertion
Ref Expression
mgcoval ((𝑉𝑋𝑊𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
Distinct variable groups:   𝐴,𝑓,𝑔,𝑥,𝑦   𝐵,𝑓,𝑔,𝑥,𝑦   𝑓,𝑉,𝑔,𝑥,𝑦   𝑓,𝑊,𝑔,𝑥,𝑦   𝑓,𝑋,𝑔,𝑥,𝑦   𝑓,𝑌,𝑔,𝑥,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑓,𝑔)   (𝑥,𝑦,𝑓,𝑔)

Proof of Theorem mgcoval
Dummy variables 𝑎 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mgc 32729 . . 3 MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ (Base‘𝑣) / 𝑎(Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))})
21a1i 11 . 2 ((𝑉𝑋𝑊𝑌) → MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ (Base‘𝑣) / 𝑎(Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))}))
3 fvexd 6917 . . 3 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) ∈ V)
4 simprl 769 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → 𝑣 = 𝑉)
54fveq2d 6906 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) = (Base‘𝑉))
6 mgcoval.1 . . . 4 𝐴 = (Base‘𝑉)
75, 6eqtr4di 2786 . . 3 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) = 𝐴)
8 fvexd 6917 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) ∈ V)
9 simplrr 776 . . . . . 6 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
109fveq2d 6906 . . . . 5 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
11 mgcoval.2 . . . . 5 𝐵 = (Base‘𝑊)
1210, 11eqtr4di 2786 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
13 simpr 483 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14 simplr 767 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴)
1513, 14oveq12d 7444 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑏m 𝑎) = (𝐵m 𝐴))
1615eleq2d 2815 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑓 ∈ (𝑏m 𝑎) ↔ 𝑓 ∈ (𝐵m 𝐴)))
1714, 13oveq12d 7444 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑎m 𝑏) = (𝐴m 𝐵))
1817eleq2d 2815 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑔 ∈ (𝑎m 𝑏) ↔ 𝑔 ∈ (𝐴m 𝐵)))
1916, 18anbi12d 630 . . . . . 6 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵))))
209adantr 479 . . . . . . . . . . . 12 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑤 = 𝑊)
2120fveq2d 6906 . . . . . . . . . . 11 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = (le‘𝑊))
22 mgcoval.4 . . . . . . . . . . 11 = (le‘𝑊)
2321, 22eqtr4di 2786 . . . . . . . . . 10 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = )
2423breqd 5163 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓𝑥)(le‘𝑤)𝑦 ↔ (𝑓𝑥) 𝑦))
254ad2antrr 724 . . . . . . . . . . . 12 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑣 = 𝑉)
2625fveq2d 6906 . . . . . . . . . . 11 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = (le‘𝑉))
27 mgcoval.3 . . . . . . . . . . 11 = (le‘𝑉)
2826, 27eqtr4di 2786 . . . . . . . . . 10 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = )
2928breqd 5163 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑥(le‘𝑣)(𝑔𝑦) ↔ 𝑥 (𝑔𝑦)))
3024, 29bibi12d 344 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3113, 30raleqbidv 3340 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ∀𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3214, 31raleqbidv 3340 . . . . . 6 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3319, 32anbi12d 630 . . . . 5 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦))) ↔ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))))
3433opabbidv 5218 . . . 4 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
358, 12, 34csbied2 3934 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
363, 7, 35csbied2 3934 . 2 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) / 𝑎(Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
37 simpl 481 . . 3 ((𝑉𝑋𝑊𝑌) → 𝑉𝑋)
3837elexd 3494 . 2 ((𝑉𝑋𝑊𝑌) → 𝑉 ∈ V)
39 simpr 483 . . 3 ((𝑉𝑋𝑊𝑌) → 𝑊𝑌)
4039elexd 3494 . 2 ((𝑉𝑋𝑊𝑌) → 𝑊 ∈ V)
41 ovexd 7461 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐵m 𝐴) ∈ V)
42 ovexd 7461 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐴m 𝐵) ∈ V)
43 simprll 777 . . 3 (((𝑉𝑋𝑊𝑌) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))) → 𝑓 ∈ (𝐵m 𝐴))
44 simprlr 778 . . 3 (((𝑉𝑋𝑊𝑌) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))) → 𝑔 ∈ (𝐴m 𝐵))
4541, 42, 43, 44opabex2 8067 . 2 ((𝑉𝑋𝑊𝑌) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))} ∈ V)
462, 36, 38, 40, 45ovmpod 7579 1 ((𝑉𝑋𝑊𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  Vcvv 3473  csb 3894   class class class wbr 5152  {copab 5214  cfv 6553  (class class class)co 7426  cmpo 7428  m cmap 8851  Basecbs 17187  lecple 17247  MGalConncmgc 32727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-mgc 32729
This theorem is referenced by:  mgcval  32735
  Copyright terms: Public domain W3C validator