Theorem mgcmnt1 30796

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 30791	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
12	4, 11	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
13	12	simplrd 769	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
14	12	simplld 767	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15	14, 3	ffvelrnd 6843	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝐵)
16	13, 15	ffvelrnd 6843	. . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑌)) ∈ 𝐴)
17		mgcmnt1.3	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
18	5, 6, 7, 8, 9, 1, 10, 4, 3	mgccole1 30794	. . 3 ⊢ (𝜑 → 𝑌 ≤ (𝐺‘(𝐹‘𝑌)))
19	5, 7	prstr 17609	. . 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 1385	. 2 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))
21	12	simprd 499	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
22		fveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
23	22	breq1d 5042	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦))
24		breq1 5035	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘𝑦)))
25	23, 24	bibi12d 349	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
26	25	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
27	26	ralbidv 3126	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
28	2, 27	rspcdv 3533	. . . 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 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → 𝑦 = (𝐹‘𝑌))
31	30	breq2d 5044	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
32	30	fveq2d 6662	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑌)))
33	32	breq2d 5044	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝑋 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌))))
34	31, 33	bibi12d 349	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ (𝐹‘𝑌) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))))
35	15, 34	rspcdv 3533	. . 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 260	1 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   class class class wbr 5032  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  lecple 16630   Proset cproset 17602  MGalConncmgc 30783

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-proset 17604  df-mgc 30785

This theorem is referenced by:  dfmgc2  30800  mgcf1olem1  30805