Theorem mgccole1 33310

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 33307	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	2, 9	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplld 779	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12		mgccole1.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
13	11, 12	ffvelcdmd 7080	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵)
14	4, 6	prsref 18349	. . 3 ⊢ ((𝑊 ∈ Proset ∧ (𝐹‘𝑋) ∈ 𝐵) → (𝐹‘𝑋) ≲ (𝐹‘𝑋))
15	1, 13, 14	syl2anc 595	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑋))
16		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
17	16	breq1d 5119	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ 𝑦))
18		breq1 5112	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘𝑦)))
19	17, 18	bibi12d 348	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
20	19	ralbidv 3188	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦))))
21	10	simprd 500	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
22	20, 21, 12	rspcdva 3582	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)))
23		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → 𝑦 = (𝐹‘𝑋))
24	23	breq2d 5121	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → ((𝐹‘𝑋) ≲ 𝑦 ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑋)))
25	23	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑋)))
26	25	breq2d 5121	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (𝑋 ≤ (𝐺‘𝑦) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋))))
27	24, 26	bibi12d 348	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑋)) → (((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))))
28	13, 27	rspcdv 3573	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑋) ≲ 𝑦 ↔ 𝑋 ≤ (𝐺‘𝑦)) → ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))))
29	22, 28	mpd 16	. 2 ⊢ (𝜑 → ((𝐹‘𝑋) ≲ (𝐹‘𝑋) ↔ 𝑋 ≤ (𝐺‘(𝐹‘𝑋))))
30	15, 29	mpbid 235	1 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312   Proset cproset 18343  MGalConncmgc 33299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-proset 18345  df-mgc 33301

This theorem is referenced by:  mgcmnt1  33312  mgcmntco  33314  dfmgc2  33316  mgcf1olem1  33321  mgcf1olem2  33322