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Theorem mgcoval 31846
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 31841 . . 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 6857 . . 3 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) ∈ V)
4 simprl 769 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → 𝑣 = 𝑉)
54fveq2d 6846 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) = (Base‘𝑉))
6 mgcoval.1 . . . 4 𝐴 = (Base‘𝑉)
75, 6eqtr4di 2794 . . 3 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) = 𝐴)
8 fvexd 6857 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) ∈ V)
9 simplrr 776 . . . . . 6 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
109fveq2d 6846 . . . . 5 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
11 mgcoval.2 . . . . 5 𝐵 = (Base‘𝑊)
1210, 11eqtr4di 2794 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
13 simpr 485 . . . . . . . . 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 7375 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑏m 𝑎) = (𝐵m 𝐴))
1615eleq2d 2823 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑓 ∈ (𝑏m 𝑎) ↔ 𝑓 ∈ (𝐵m 𝐴)))
1714, 13oveq12d 7375 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑎m 𝑏) = (𝐴m 𝐵))
1817eleq2d 2823 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑔 ∈ (𝑎m 𝑏) ↔ 𝑔 ∈ (𝐴m 𝐵)))
1916, 18anbi12d 631 . . . . . 6 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵))))
209adantr 481 . . . . . . . . . . . 12 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑤 = 𝑊)
2120fveq2d 6846 . . . . . . . . . . 11 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = (le‘𝑊))
22 mgcoval.4 . . . . . . . . . . 11 = (le‘𝑊)
2321, 22eqtr4di 2794 . . . . . . . . . 10 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = )
2423breqd 5116 . . . . . . . . 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 6846 . . . . . . . . . . 11 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = (le‘𝑉))
27 mgcoval.3 . . . . . . . . . . 11 = (le‘𝑉)
2826, 27eqtr4di 2794 . . . . . . . . . 10 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = )
2928breqd 5116 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑥(le‘𝑣)(𝑔𝑦) ↔ 𝑥 (𝑔𝑦)))
3024, 29bibi12d 345 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3113, 30raleqbidv 3319 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ∀𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3214, 31raleqbidv 3319 . . . . . 6 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3319, 32anbi12d 631 . . . . 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 5171 . . . 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 3895 . . 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 3895 . 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 483 . . 3 ((𝑉𝑋𝑊𝑌) → 𝑉𝑋)
3837elexd 3465 . 2 ((𝑉𝑋𝑊𝑌) → 𝑉 ∈ V)
39 simpr 485 . . 3 ((𝑉𝑋𝑊𝑌) → 𝑊𝑌)
4039elexd 3465 . 2 ((𝑉𝑋𝑊𝑌) → 𝑊 ∈ V)
41 ovexd 7392 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐵m 𝐴) ∈ V)
42 ovexd 7392 . . 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 7989 . 2 ((𝑉𝑋𝑊𝑌) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))} ∈ V)
462, 36, 38, 40, 45ovmpod 7507 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 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  csb 3855   class class class wbr 5105  {copab 5167  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  Basecbs 17083  lecple 17140  MGalConncmgc 31839
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-mgc 31841
This theorem is referenced by:  mgcval  31847
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