dfmgc2 - Mathbox for Thierry Arnoux

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Theorem dfmgc2 33257

Description: Alternate definition of the monotone Galois connection. (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 )

**Assertion**
Ref	Expression
dfmgc2	⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))))

Distinct variable groups: 𝑣, ≤ 𝑣, ≲ 𝑣,𝐴,𝑥,𝑦 𝑣,𝐵,𝑥,𝑦 𝑣,𝑉,𝑥,𝑦 𝑣,𝑊,𝑥,𝑦 𝑥,𝐹,𝑦 𝑥,𝐺,𝑦 𝑢, ≤ 𝜑,𝑥,𝑦 𝑥,𝐻,𝑦 𝜑,𝑢,𝑣 𝑢,𝐻,𝑣 𝑢,𝐺,𝑣 𝑢,𝐹,𝑣 𝑢,𝐵 𝑢, ≲ 𝑥, ≤ ,𝑦 𝑥, ≲ ,𝑦 Allowed substitution hints: 𝐴(𝑢) 𝑉(𝑢) 𝑊(𝑢)

Proof of Theorem dfmgc2 Dummy variables 𝑖 𝑗 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgcoval.1	. . . . 5 ⊢ 𝐴 = (Base‘𝑉)
2		mgcoval.2	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		mgcoval.3	. . . . 5 ⊢ ≤ = (le‘𝑉)
4		mgcoval.4	. . . . 5 ⊢ ≲ = (le‘𝑊)
5		mgcval.1	. . . . 5 ⊢ 𝐻 = (𝑉MGalConn𝑊)
6		mgcval.2	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ Proset )
7		mgcval.3	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Proset )
8	1, 2, 3, 4, 5, 6, 7	mgcval 33248	. . . 4 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
9	8	simprbda 503	. . 3 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴))
10	6	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑉 ∈ Proset )
11	7	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑊 ∈ Proset )
12		simp-4r 795	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝐹𝐻𝐺)
13		simpllr 787	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑥 ∈ 𝐴)
14		simplr 780	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ 𝐴)
15		simpr 489	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
16	1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15	mgcmnt1 33253	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → (𝐹‘𝑥) ≲ (𝐹‘𝑦))
17	16	ex 417	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
18	17	anasss 471	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
19	18	ralrimivva 3214	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
20	6	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑉 ∈ Proset )
21	7	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑊 ∈ Proset )
22		simp-4r 795	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝐹𝐻𝐺)
23		simpllr 787	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑢 ∈ 𝐵)
24		simplr 780	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑣 ∈ 𝐵)
25		simpr 489	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑢 ≲ 𝑣)
26	1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25	mgcmnt2 33254	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → (𝐺‘𝑢) ≤ (𝐺‘𝑣))
27	26	ex 417	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
28	27	anasss 471	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
29	28	ralrimivva 3214	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
30	19, 29	jca 520	. . . 4 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))))
31	6	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑉 ∈ Proset )
32	7	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑊 ∈ Proset )
33		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝐹𝐻𝐺)
34		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
35	1, 2, 3, 4, 5, 31, 32, 33, 34	mgccole2 33252	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)
36	35	ralrimiva 3163	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)
37	6	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑉 ∈ Proset )
38	7	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑊 ∈ Proset )
39		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝐹𝐻𝐺)
40		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
41	1, 2, 3, 4, 5, 37, 38, 39, 40	mgccole1 33251	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
42	41	ralrimiva 3163	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
43	36, 42	jca 520	. . . 4 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))
44	30, 43	jca 520	. . 3 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))
45	9, 44	jca 520	. 2 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))))
46	6	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → 𝑉 ∈ Proset )
47	7	ad4antr 744	. . . . . 6 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → 𝑊 ∈ Proset )
48		simp-4r 795	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴))
49	48	simpld 499	. . . . . 6 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → 𝐹:𝐴⟶𝐵)
50	48	simprd 500	. . . . . 6 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → 𝐺:𝐵⟶𝐴)
51		simpllr 787	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))))
52	51	simpld 499	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
53		breq1 5116	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → (𝑥 ≤ 𝑦 ↔ 𝑚 ≤ 𝑦))
54		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
55	54	breq1d 5123	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑚) ≲ (𝐹‘𝑦)))
56	53, 55	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑚 ≤ 𝑦 → (𝐹‘𝑚) ≲ (𝐹‘𝑦))))
57		breq2 5117	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑚 ≤ 𝑦 ↔ 𝑚 ≤ 𝑛))
58		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
59	58	breq2d 5125	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → ((𝐹‘𝑚) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑚) ≲ (𝐹‘𝑛)))
60	57, 59	imbi12d 347	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝑚 ≤ 𝑦 → (𝐹‘𝑚) ≲ (𝐹‘𝑦)) ↔ (𝑚 ≤ 𝑛 → (𝐹‘𝑚) ≲ (𝐹‘𝑛))))
61	56, 60	cbvral2vw 3253	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ ∀𝑚 ∈ 𝐴 ∀𝑛 ∈ 𝐴 (𝑚 ≤ 𝑛 → (𝐹‘𝑚) ≲ (𝐹‘𝑛)))
62	52, 61	sylib 221	. . . . . 6 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → ∀𝑚 ∈ 𝐴 ∀𝑛 ∈ 𝐴 (𝑚 ≤ 𝑛 → (𝐹‘𝑚) ≲ (𝐹‘𝑛)))
63	51	simprd 500	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
64		breq1 5116	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (𝑢 ≲ 𝑣 ↔ 𝑖 ≲ 𝑣))
65		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑢 = 𝑖 → (𝐺‘𝑢) = (𝐺‘𝑖))
66	65	breq1d 5123	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → ((𝐺‘𝑢) ≤ (𝐺‘𝑣) ↔ (𝐺‘𝑖) ≤ (𝐺‘𝑣)))
67	64, 66	imbi12d 347	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → ((𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)) ↔ (𝑖 ≲ 𝑣 → (𝐺‘𝑖) ≤ (𝐺‘𝑣))))
68		breq2 5117	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → (𝑖 ≲ 𝑣 ↔ 𝑖 ≲ 𝑗))
69		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗))
70	69	breq2d 5125	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → ((𝐺‘𝑖) ≤ (𝐺‘𝑣) ↔ (𝐺‘𝑖) ≤ (𝐺‘𝑗)))
71	68, 70	imbi12d 347	. . . . . . . 8 ⊢ (𝑣 = 𝑗 → ((𝑖 ≲ 𝑣 → (𝐺‘𝑖) ≤ (𝐺‘𝑣)) ↔ (𝑖 ≲ 𝑗 → (𝐺‘𝑖) ≤ (𝐺‘𝑗))))
72	67, 71	cbvral2vw 3253	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)) ↔ ∀𝑖 ∈ 𝐵 ∀𝑗 ∈ 𝐵 (𝑖 ≲ 𝑗 → (𝐺‘𝑖) ≤ (𝐺‘𝑗)))
73	63, 72	sylib 221	. . . . . 6 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → ∀𝑖 ∈ 𝐵 ∀𝑗 ∈ 𝐵 (𝑖 ≲ 𝑗 → (𝐺‘𝑖) ≤ (𝐺‘𝑗)))
74		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → 𝑥 = 𝑚)
75		2fveq3 6887	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝐺‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑚)))
76	74, 75	breq12d 5126	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑥 ≤ (𝐺‘(𝐹‘𝑥)) ↔ 𝑚 ≤ (𝐺‘(𝐹‘𝑚))))
77		simplr 780	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) ∧ 𝑚 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
78		simpr 489	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐴)
79	76, 77, 78	rspcdva 3591	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) ∧ 𝑚 ∈ 𝐴) → 𝑚 ≤ (𝐺‘(𝐹‘𝑚)))
80		2fveq3 6887	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝑖)))
81		id 23	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → 𝑢 = 𝑖)
82	80, 81	breq12d 5126	. . . . . . 7 ⊢ (𝑢 = 𝑖 → ((𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ↔ (𝐹‘(𝐺‘𝑖)) ≲ 𝑖))
83		simpllr 787	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) ∧ 𝑖 ∈ 𝐵) → ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)
84		simpr 489	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ 𝐵)
85	82, 83, 84	rspcdva 3591	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) ∧ 𝑖 ∈ 𝐵) → (𝐹‘(𝐺‘𝑖)) ≲ 𝑖)
86	1, 2, 3, 4, 5, 46, 47, 49, 50, 62, 73, 79, 85	dfmgc2lem 33256	. . . . 5 ⊢ (((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) → 𝐹𝐻𝐺)
87	86	anasss 471	. . . 4 ⊢ ((((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))) → 𝐹𝐻𝐺)
88	87	anasss 471	. . 3 ⊢ (((𝜑 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))) → 𝐹𝐻𝐺)
89	88	anasss 471	. 2 ⊢ ((𝜑 ∧ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))) → 𝐹𝐻𝐺)
90	45, 89	impbida 812	1 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 class class class wbr 5113 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 Basecbs 17269 lecple 17317 Proset cproset 18348 MGalConncmgc 33240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8826 df-proset 18350 df-mgc 33242

This theorem is referenced by: mgcmnt1d 33258 mgcmnt2d 33259 mgcf1olem1 33262 mgcf1olem2 33263 mgcf1o 33264

W3C validator