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Theorem mgcmnt1 32149
Description: The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
Hypotheses
Ref Expression
mgcoval.1 𝐴 = (Base‘𝑉)
mgcoval.2 𝐵 = (Base‘𝑊)
mgcoval.3 = (le‘𝑉)
mgcoval.4 = (le‘𝑊)
mgcval.1 𝐻 = (𝑉MGalConn𝑊)
mgcval.2 (𝜑𝑉 ∈ Proset )
mgcval.3 (𝜑𝑊 ∈ Proset )
mgccole.1 (𝜑𝐹𝐻𝐺)
mgcmnt1.1 (𝜑𝑋𝐴)
mgcmnt1.2 (𝜑𝑌𝐴)
mgcmnt1.3 (𝜑𝑋 𝑌)
Assertion
Ref Expression
mgcmnt1 (𝜑 → (𝐹𝑋) (𝐹𝑌))

Proof of Theorem mgcmnt1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.2 . . 3 (𝜑𝑉 ∈ Proset )
2 mgcmnt1.1 . . 3 (𝜑𝑋𝐴)
3 mgcmnt1.2 . . 3 (𝜑𝑌𝐴)
4 mgccole.1 . . . . . 6 (𝜑𝐹𝐻𝐺)
5 mgcoval.1 . . . . . . 7 𝐴 = (Base‘𝑉)
6 mgcoval.2 . . . . . . 7 𝐵 = (Base‘𝑊)
7 mgcoval.3 . . . . . . 7 = (le‘𝑉)
8 mgcoval.4 . . . . . . 7 = (le‘𝑊)
9 mgcval.1 . . . . . . 7 𝐻 = (𝑉MGalConn𝑊)
10 mgcval.3 . . . . . . 7 (𝜑𝑊 ∈ Proset )
115, 6, 7, 8, 9, 1, 10mgcval 32144 . . . . . 6 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
124, 11mpbid 231 . . . . 5 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
1312simplrd 768 . . . 4 (𝜑𝐺:𝐵𝐴)
1412simplld 766 . . . . 5 (𝜑𝐹:𝐴𝐵)
1514, 3ffvelcdmd 7084 . . . 4 (𝜑 → (𝐹𝑌) ∈ 𝐵)
1613, 15ffvelcdmd 7084 . . 3 (𝜑 → (𝐺‘(𝐹𝑌)) ∈ 𝐴)
17 mgcmnt1.3 . . 3 (𝜑𝑋 𝑌)
185, 6, 7, 8, 9, 1, 10, 4, 3mgccole1 32147 . . 3 (𝜑𝑌 (𝐺‘(𝐹𝑌)))
195, 7prstr 18249 . . 3 ((𝑉 ∈ Proset ∧ (𝑋𝐴𝑌𝐴 ∧ (𝐺‘(𝐹𝑌)) ∈ 𝐴) ∧ (𝑋 𝑌𝑌 (𝐺‘(𝐹𝑌)))) → 𝑋 (𝐺‘(𝐹𝑌)))
201, 2, 3, 16, 17, 18, 19syl132anc 1388 . 2 (𝜑𝑋 (𝐺‘(𝐹𝑌)))
2112simprd 496 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))
22 fveq2 6888 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2322breq1d 5157 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) 𝑦 ↔ (𝐹𝑋) 𝑦))
24 breq1 5150 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 (𝐺𝑦) ↔ 𝑋 (𝐺𝑦)))
2523, 24bibi12d 345 . . . . . . 7 (𝑥 = 𝑋 → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2625adantl 482 . . . . . 6 ((𝜑𝑥 = 𝑋) → (((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2726ralbidv 3177 . . . . 5 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) ↔ ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
282, 27rspcdv 3604 . . . 4 (𝜑 → (∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)) → ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦))))
2921, 28mpd 15 . . 3 (𝜑 → ∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)))
30 simpr 485 . . . . . 6 ((𝜑𝑦 = (𝐹𝑌)) → 𝑦 = (𝐹𝑌))
3130breq2d 5159 . . . . 5 ((𝜑𝑦 = (𝐹𝑌)) → ((𝐹𝑋) 𝑦 ↔ (𝐹𝑋) (𝐹𝑌)))
3230fveq2d 6892 . . . . . 6 ((𝜑𝑦 = (𝐹𝑌)) → (𝐺𝑦) = (𝐺‘(𝐹𝑌)))
3332breq2d 5159 . . . . 5 ((𝜑𝑦 = (𝐹𝑌)) → (𝑋 (𝐺𝑦) ↔ 𝑋 (𝐺‘(𝐹𝑌))))
3431, 33bibi12d 345 . . . 4 ((𝜑𝑦 = (𝐹𝑌)) → (((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) ↔ ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌)))))
3515, 34rspcdv 3604 . . 3 (𝜑 → (∀𝑦𝐵 ((𝐹𝑋) 𝑦𝑋 (𝐺𝑦)) → ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌)))))
3629, 35mpd 15 . 2 (𝜑 → ((𝐹𝑋) (𝐹𝑌) ↔ 𝑋 (𝐺‘(𝐹𝑌))))
3720, 36mpbird 256 1 (𝜑 → (𝐹𝑋) (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061   class class class wbr 5147  wf 6536  cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200   Proset cproset 18242  MGalConncmgc 32136
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-proset 18244  df-mgc 32138
This theorem is referenced by:  dfmgc2  32153  mgcf1olem1  32158
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