Theorem dfmgc2 31856

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, ≤ ,𝑣   𝑥, ≤ ,𝑦   𝑢, ≲   𝑥, ≲ ,𝑦   𝑢,𝐵   𝑢,𝐹,𝑣   𝑢,𝐺,𝑣   𝑢,𝐻,𝑣   𝑥,𝐻,𝑦   𝜑,𝑢,𝑣   𝜑,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 31847	. . . 4 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
9	8	simprbda 499	. . 3 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴))
10	6	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑉 ∈ Proset )
11	7	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑊 ∈ Proset )
12		simp-4r 782	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝐹𝐻𝐺)
13		simpllr 774	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑥 ∈ 𝐴)
14		simplr 767	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ 𝐴)
15		simpr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
16	1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15	mgcmnt1 31852	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → (𝐹‘𝑥) ≲ (𝐹‘𝑦))
17	16	ex 413	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
18	17	anasss 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
19	18	ralrimivva 3197	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
20	6	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑉 ∈ Proset )
21	7	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑊 ∈ Proset )
22		simp-4r 782	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝐹𝐻𝐺)
23		simpllr 774	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑢 ∈ 𝐵)
24		simplr 767	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑣 ∈ 𝐵)
25		simpr 485	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑢 ≲ 𝑣)
26	1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25	mgcmnt2 31853	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → (𝐺‘𝑢) ≤ (𝐺‘𝑣))
27	26	ex 413	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
28	27	anasss 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
29	28	ralrimivva 3197	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
30	19, 29	jca 512	. . . 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑉 ∈ Proset )
32	7	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑊 ∈ Proset )
33		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝐹𝐻𝐺)
34		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
35	1, 2, 3, 4, 5, 31, 32, 33, 34	mgccole2 31851	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)
36	35	ralrimiva 3143	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)
37	6	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑉 ∈ Proset )
38	7	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑊 ∈ Proset )
39		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝐹𝐻𝐺)
40		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
41	1, 2, 3, 4, 5, 37, 38, 39, 40	mgccole1 31850	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
42	41	ralrimiva 3143	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
43	36, 42	jca 512	. . . 4 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))
44	30, 43	jca 512	. . 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 512	. 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 730	. . . . . 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 730	. . . . . 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 782	. . . . . . 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 495	. . . . . 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 496	. . . . . 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 774	. . . . . . . 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 495	. . . . . . 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 5108	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → (𝑥 ≤ 𝑦 ↔ 𝑚 ≤ 𝑦))
54		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
55	54	breq1d 5115	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑚) ≲ (𝐹‘𝑦)))
56	53, 55	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑚 ≤ 𝑦 → (𝐹‘𝑚) ≲ (𝐹‘𝑦))))
57		breq2 5109	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑚 ≤ 𝑦 ↔ 𝑚 ≤ 𝑛))
58		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
59	58	breq2d 5117	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → ((𝐹‘𝑚) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑚) ≲ (𝐹‘𝑛)))
60	57, 59	imbi12d 344	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝑚 ≤ 𝑦 → (𝐹‘𝑚) ≲ (𝐹‘𝑦)) ↔ (𝑚 ≤ 𝑛 → (𝐹‘𝑚) ≲ (𝐹‘𝑛))))
61	56, 60	cbvral2vw 3227	. . . . . . 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 217	. . . . . 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 496	. . . . . . 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 5108	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (𝑢 ≲ 𝑣 ↔ 𝑖 ≲ 𝑣))
65		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑢 = 𝑖 → (𝐺‘𝑢) = (𝐺‘𝑖))
66	65	breq1d 5115	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → ((𝐺‘𝑢) ≤ (𝐺‘𝑣) ↔ (𝐺‘𝑖) ≤ (𝐺‘𝑣)))
67	64, 66	imbi12d 344	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → ((𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)) ↔ (𝑖 ≲ 𝑣 → (𝐺‘𝑖) ≤ (𝐺‘𝑣))))
68		breq2 5109	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → (𝑖 ≲ 𝑣 ↔ 𝑖 ≲ 𝑗))
69		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗))
70	69	breq2d 5117	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → ((𝐺‘𝑖) ≤ (𝐺‘𝑣) ↔ (𝐺‘𝑖) ≤ (𝐺‘𝑗)))
71	68, 70	imbi12d 344	. . . . . . . 8 ⊢ (𝑣 = 𝑗 → ((𝑖 ≲ 𝑣 → (𝐺‘𝑖) ≤ (𝐺‘𝑣)) ↔ (𝑖 ≲ 𝑗 → (𝐺‘𝑖) ≤ (𝐺‘𝑗))))
72	67, 71	cbvral2vw 3227	. . . . . . 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 217	. . . . . 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 22	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → 𝑥 = 𝑚)
75		2fveq3 6847	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝐺‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑚)))
76	74, 75	breq12d 5118	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑥 ≤ (𝐺‘(𝐹‘𝑥)) ↔ 𝑚 ≤ (𝐺‘(𝐹‘𝑚))))
77		simplr 767	. . . . . . 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 485	. . . . . . 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 3582	. . . . . 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 6847	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝑖)))
81		id 22	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → 𝑢 = 𝑖)
82	80, 81	breq12d 5118	. . . . . . 7 ⊢ (𝑢 = 𝑖 → ((𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ↔ (𝐹‘(𝐺‘𝑖)) ≲ 𝑖))
83		simpllr 774	. . . . . . 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 485	. . . . . . 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 3582	. . . . . 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 31855	. . . . 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 467	. . . 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 467	. . 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 467	. 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 799	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)))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064   class class class wbr 5105  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140   Proset cproset 18182  MGalConncmgc 31839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-proset 18184  df-mgc 31841

This theorem is referenced by:  mgcmnt1d  31857  mgcmnt2d  31858  mgcf1olem1  31861  mgcf1olem2  31862  mgcf1o  31863