Theorem mgcmnt1 32925

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 32920	. . . . . 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 769	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
14	12	simplld 767	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15	14, 3	ffvelcdmd 7060	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝐵)
16	13, 15	ffvelcdmd 7060	. . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑌)) ∈ 𝐴)
17		mgcmnt1.3	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
18	5, 6, 7, 8, 9, 1, 10, 4, 3	mgccole1 32923	. . 3 ⊢ (𝜑 → 𝑌 ≤ (𝐺‘(𝐹‘𝑌)))
19	5, 7	prstr 18267	. . 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 6861	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
23	22	breq1d 5120	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦))
24		breq1 5113	. . . . . . . 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 3157	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
28	2, 27	rspcdv 3583	. . . 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 5122	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
32	30	fveq2d 6865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑌)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑌)))
33	32	breq2d 5122	. . . . 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 3583	. . 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 2109  ∀wral 3045   class class class wbr 5110  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234   Proset cproset 18260  MGalConncmgc 32912

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-proset 18262  df-mgc 32914

This theorem is referenced by:  dfmgc2  32929  mgcf1olem1  32934