dfmgc2 - Mathbox for Thierry Arnoux

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Theorem dfmgc2 32971

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 32962	. . . 4 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
9	8	simprbda 498	. . 3 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴))
10	6	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑉 ∈ Proset )
11	7	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑊 ∈ Proset )
12		simp-4r 784	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝐹𝐻𝐺)
13		simpllr 776	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑥 ∈ 𝐴)
14		simplr 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ 𝐴)
15		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦)
16	1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15	mgcmnt1 32967	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≤ 𝑦) → (𝐹‘𝑥) ≲ (𝐹‘𝑦))
17	16	ex 412	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
18	17	anasss 466	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
19	18	ralrimivva 3200	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
20	6	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑉 ∈ Proset )
21	7	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑊 ∈ Proset )
22		simp-4r 784	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝐹𝐻𝐺)
23		simpllr 776	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑢 ∈ 𝐵)
24		simplr 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑣 ∈ 𝐵)
25		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → 𝑢 ≲ 𝑣)
26	1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25	mgcmnt2 32968	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ≲ 𝑣) → (𝐺‘𝑢) ≤ (𝐺‘𝑣))
27	26	ex 412	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
28	27	anasss 466	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
29	28	ralrimivva 3200	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
30	19, 29	jca 511	. . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑉 ∈ Proset )
32	7	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑊 ∈ Proset )
33		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝐹𝐻𝐺)
34		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
35	1, 2, 3, 4, 5, 31, 32, 33, 34	mgccole2 32966	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)
36	35	ralrimiva 3144	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)
37	6	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑉 ∈ Proset )
38	7	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑊 ∈ Proset )
39		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝐹𝐻𝐺)
40		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
41	1, 2, 3, 4, 5, 37, 38, 39, 40	mgccole1 32965	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹𝐻𝐺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
42	41	ralrimiva 3144	. . . . 5 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))
43	36, 42	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝐹𝐻𝐺) → (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))
44	30, 43	jca 511	. . 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 511	. 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 732	. . . . . 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 732	. . . . . 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 784	. . . . . . 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 494	. . . . . 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 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))) → 𝐺:𝐵⟶𝐴)
51		simpllr 776	. . . . . . . 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 494	. . . . . . 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 5151	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → (𝑥 ≤ 𝑦 ↔ 𝑚 ≤ 𝑦))
54		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
55	54	breq1d 5158	. . . . . . . . 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 5152	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑚 ≤ 𝑦 ↔ 𝑚 ≤ 𝑛))
58		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
59	58	breq2d 5160	. . . . . . . . 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 3239	. . . . . . 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 218	. . . . . 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 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))) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))
64		breq1 5151	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (𝑢 ≲ 𝑣 ↔ 𝑖 ≲ 𝑣))
65		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑢 = 𝑖 → (𝐺‘𝑢) = (𝐺‘𝑖))
66	65	breq1d 5158	. . . . . . . . 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 5152	. . . . . . . . 9 ⊢ (𝑣 = 𝑗 → (𝑖 ≲ 𝑣 ↔ 𝑖 ≲ 𝑗))
69		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑣 = 𝑗 → (𝐺‘𝑣) = (𝐺‘𝑗))
70	69	breq2d 5160	. . . . . . . . 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 3239	. . . . . . 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 218	. . . . . 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 6912	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝐺‘(𝐹‘𝑥)) = (𝐺‘(𝐹‘𝑚)))
76	74, 75	breq12d 5161	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑥 ≤ (𝐺‘(𝐹‘𝑥)) ↔ 𝑚 ≤ (𝐺‘(𝐹‘𝑚))))
77		simplr 769	. . . . . . 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 484	. . . . . . 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 3623	. . . . . 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 6912	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝑖)))
81		id 22	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → 𝑢 = 𝑖)
82	80, 81	breq12d 5161	. . . . . . 7 ⊢ (𝑢 = 𝑖 → ((𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ↔ (𝐹‘(𝐺‘𝑖)) ≲ 𝑖))
83		simpllr 776	. . . . . . 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 484	. . . . . . 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 3623	. . . . . 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 32970	. . . . 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 466	. . . 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 466	. . 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 466	. 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 801	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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 Proset cproset 18350 MGalConncmgc 32954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-proset 18352 df-mgc 32956

This theorem is referenced by: mgcmnt1d 32972 mgcmnt2d 32973 mgcf1olem1 32976 mgcf1olem2 32977 mgcf1o 32978

W3C validator