Theorem mgcmnt1 32664

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 32659	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
12	4, 11	mpbid 231	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
13	12	simplrd 767	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
14	12	simplld 765	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15	14, 3	ffvelcdmd 7080	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝐵)
16	13, 15	ffvelcdmd 7080	. . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑌)) ∈ 𝐴)
17		mgcmnt1.3	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
18	5, 6, 7, 8, 9, 1, 10, 4, 3	mgccole1 32662	. . 3 ⊢ (𝜑 → 𝑌 ≤ (𝐺‘(𝐹‘𝑌)))
19	5, 7	prstr 18262	. . 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 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
22		fveq2 6884	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
23	22	breq1d 5151	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦))
24		breq1 5144	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘𝑦)))
25	23, 24	bibi12d 345	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
26	25	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
27	26	ralbidv 3171	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
28	2, 27	rspcdv 3598	. . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → 𝑦 = (𝐹‘𝑌))
31	30	breq2d 5153	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
32	30	fveq2d 6888	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑌)))
33	32	breq2d 5153	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝑋 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌))))
34	31, 33	bibi12d 345	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ (𝐹‘𝑌) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))))
35	15, 34	rspcdv 3598	. . 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 257	1 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055   class class class wbr 5141  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210   Proset cproset 18255  MGalConncmgc 32651

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-proset 18257  df-mgc 32653

This theorem is referenced by:  dfmgc2  32668  mgcf1olem1  32673