Theorem mgcmnt1 33072

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	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcval.3	⊢ (𝜑 → 𝑊 ∈ Proset )
mgccole.1	⊢ (𝜑 → 𝐹𝐻𝐺)
mgcmnt1.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
mgcmnt1.2	⊢ (𝜑 → 𝑌 ∈ 𝐴)
mgcmnt1.3	⊢ (𝜑 → 𝑋 ≤ 𝑌)

Assertion

Ref	Expression
mgcmnt1	⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))

Proof of Theorem mgcmnt1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgcval.2	. . 3 ⊢ (𝜑 → 𝑉 ∈ Proset )
2		mgcmnt1.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
3		mgcmnt1.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
4		mgccole.1	. . . . . 6 ⊢ (𝜑 → 𝐹𝐻𝐺)
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 ⊢ (𝜑 → 𝑊 ∈ Proset )
11	5, 6, 7, 8, 9, 1, 10	mgcval 33067	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
12	4, 11	mpbid 233	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
13	12	simplrd 775	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
14	12	simplld 773	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15	14, 3	ffvelcdmd 7027	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝐵)
16	13, 15	ffvelcdmd 7027	. . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑌)) ∈ 𝐴)
17		mgcmnt1.3	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
18	5, 6, 7, 8, 9, 1, 10, 4, 3	mgccole1 33070	. . 3 ⊢ (𝜑 → 𝑌 ≤ (𝐺‘(𝐹‘𝑌)))
19	5, 7	prstr 18257	. . 3 ⊢ ((𝑉 ∈ Proset ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ (𝐺‘(𝐹‘𝑌)) ∈ 𝐴) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ (𝐺‘(𝐹‘𝑌)))) → 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))
20	1, 2, 3, 16, 17, 18, 19	syl132anc 1396	. 2 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))
21	12	simprd 496	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
22		fveq2 6828	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
23	22	breq1d 5083	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦))
24		breq1 5076	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘𝑦)))
25	23, 24	bibi12d 346	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
26	25	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
27	26	ralbidv 3162	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
28	2, 27	rspcdv 3552	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
29	21, 28	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)))
30		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → 𝑦 = (𝐹‘𝑌))
31	30	breq2d 5085	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
32	30	fveq2d 6832	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑌)))
33	32	breq2d 5085	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝑋 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌))))
34	31, 33	bibi12d 346	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ (𝐹‘𝑌) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))))
35	15, 34	rspcdv 3552	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) → ((𝐹‘𝑋) ≲ (𝐹‘𝑌) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))))
36	29, 35	mpd 15	. 2 ⊢ (𝜑 → ((𝐹‘𝑋) ≲ (𝐹‘𝑌) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌))))
37	20, 36	mpbird 258	1 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   class class class wbr 5073  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  Basecbs 17171  lecple 17219   Proset cproset 18250  MGalConncmgc 33059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8766  df-proset 18252  df-mgc 33061

This theorem is referenced by:  dfmgc2  33076  mgcf1olem1  33081