Theorem mgcmnt1 32982

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 32977	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
12	4, 11	mpbid 232	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
13	12	simplrd 770	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
14	12	simplld 768	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15	14, 3	ffvelcdmd 7105	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝐵)
16	13, 15	ffvelcdmd 7105	. . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑌)) ∈ 𝐴)
17		mgcmnt1.3	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
18	5, 6, 7, 8, 9, 1, 10, 4, 3	mgccole1 32980	. . 3 ⊢ (𝜑 → 𝑌 ≤ (𝐺‘(𝐹‘𝑌)))
19	5, 7	prstr 18345	. . 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 1390	. 2 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑌)))
21	12	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
22		fveq2 6906	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
23	22	breq1d 5153	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦))
24		breq1 5146	. . . . . . . 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 3178	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
28	2, 27	rspcdv 3614	. . . 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 5155	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
32	30	fveq2d 6910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑌)))
33	32	breq2d 5155	. . . . 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 3614	. . 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   class class class wbr 5143  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304   Proset cproset 18338  MGalConncmgc 32969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-proset 18340  df-mgc 32971

This theorem is referenced by:  dfmgc2  32986  mgcf1olem1  32991