Theorem mgccole1 33072

Description: An inequality for the kernel operator 𝐺 ∘ 𝐹. (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)
mgccole1.2	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
mgccole1	⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))

Proof of Theorem mgccole1

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

Step	Hyp	Ref	Expression
1		mgcval.3	. . 3 ⊢ (𝜑 → 𝑊 ∈ Proset )
2		mgccole.1	. . . . . 6 ⊢ (𝜑 → 𝐹𝐻𝐺)
3		mgcoval.1	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑉)
4		mgcoval.2	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
5		mgcoval.3	. . . . . . 7 ⊢ ≤ = (le‘𝑉)
6		mgcoval.4	. . . . . . 7 ⊢ ≲ = (le‘𝑊)
7		mgcval.1	. . . . . . 7 ⊢ 𝐻 = (𝑉MGalConn𝑊)
8		mgcval.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Proset )
9	3, 4, 5, 6, 7, 8, 1	mgcval 33069	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	2, 9	mpbid 232	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplld 767	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12		mgccole1.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
13	11, 12	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵)
14	4, 6	prsref 18221	. . 3 ⊢ ((𝑊 ∈ Proset ∧ (𝐹‘𝑋) ∈ 𝐵) → (𝐹‘𝑋) ≲ (𝐹‘𝑋))
15	1, 13, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑋))
16		fveq2 6834	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
17	16	breq1d 5108	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦))
18		breq1 5101	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘𝑦)))
19	17, 18	bibi12d 345	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
20	19	ralbidv 3159	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
21	10	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
22	20, 21, 12	rspcdva 3577	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)))
23		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → 𝑦 = (𝐹‘𝑋))
24	23	breq2d 5110	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑋)))
25	23	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑋)))
26	25	breq2d 5110	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝑋 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋))))
27	24, 26	bibi12d 345	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))))
28	13, 27	rspcdv 3568	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) → ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))))
29	22, 28	mpd 15	. 2 ⊢ (𝜑 → ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋))))
30	15, 29	mpbid 232	1 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   class class class wbr 5098  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184   Proset cproset 18215  MGalConncmgc 33061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-proset 18217  df-mgc 33063

This theorem is referenced by:  mgcmnt1  33074  mgcmntco  33076  dfmgc2  33078  mgcf1olem1  33083  mgcf1olem2  33084