Step | Hyp | Ref
| Expression |
1 | | cauappcvgpr.f |
. . . . . . 7
⊢ (φ → 𝐹:Q⟶Q) |
2 | | cauappcvgpr.app |
. . . . . . 7
⊢ (φ → ∀𝑝 ∈
Q ∀𝑞 ∈
Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) |
3 | | cauappcvgpr.bnd |
. . . . . . 7
⊢ (φ → ∀𝑝 ∈
Q A
<Q (𝐹‘𝑝)) |
4 | | cauappcvgpr.lim |
. . . . . . 7
⊢ 𝐿 = 〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q (𝐹‘𝑞)}, {u
∈ Q ∣ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q u}〉 |
5 | 1, 2, 3, 4 | cauappcvgprlemcl 6625 |
. . . . . 6
⊢ (φ → 𝐿 ∈
P) |
6 | | cauappcvgprlemladd.s |
. . . . . . 7
⊢ (φ → 𝑆 ∈
Q) |
7 | | nqprlu 6530 |
. . . . . . 7
⊢ (𝑆 ∈ Q → 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉 ∈
P) |
8 | 6, 7 | syl 14 |
. . . . . 6
⊢ (φ → 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉 ∈
P) |
9 | | df-iplp 6451 |
. . . . . . 7
⊢
+P = (x ∈ P, y ∈
P ↦ 〈{f ∈ Q ∣ ∃g ∈ Q ∃ℎ
∈ Q (g ∈
(1st ‘x) ∧ ℎ
∈ (1st ‘y) ∧ f = (g
+Q ℎ))}, {f
∈ Q ∣ ∃g ∈ Q ∃ℎ
∈ Q (g ∈
(2nd ‘x) ∧ ℎ
∈ (2nd ‘y) ∧ f = (g
+Q ℎ))}〉) |
10 | | addclnq 6359 |
. . . . . . 7
⊢
((g ∈ Q ∧ ℎ
∈ Q) → (g +Q ℎ) ∈
Q) |
11 | 9, 10 | genpelvu 6496 |
. . . . . 6
⊢ ((𝐿 ∈ P ∧ 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉 ∈
P) → (𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)) ↔ ∃𝑠 ∈
(2nd ‘𝐿)∃𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)𝑟 = (𝑠 +Q 𝑡))) |
12 | 5, 8, 11 | syl2anc 391 |
. . . . 5
⊢ (φ → (𝑟 ∈
(2nd ‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)) ↔ ∃𝑠 ∈
(2nd ‘𝐿)∃𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)𝑟 = (𝑠 +Q 𝑡))) |
13 | 12 | biimpa 280 |
. . . 4
⊢ ((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) → ∃𝑠 ∈
(2nd ‘𝐿)∃𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)𝑟 = (𝑠 +Q 𝑡)) |
14 | | breq2 3759 |
. . . . . . . . . . . . . . . 16
⊢ (u = 𝑠 → (((𝐹‘𝑞) +Q 𝑞) <Q u ↔ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
15 | 14 | rexbidv 2321 |
. . . . . . . . . . . . . . 15
⊢ (u = 𝑠 → (∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q u ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
16 | 4 | fveq2i 5124 |
. . . . . . . . . . . . . . . 16
⊢
(2nd ‘𝐿) = (2nd ‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q (𝐹‘𝑞)}, {u
∈ Q ∣ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q u}〉) |
17 | | nqex 6347 |
. . . . . . . . . . . . . . . . . 18
⊢
Q ∈ V |
18 | 17 | rabex 3892 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q (𝐹‘𝑞)} ∈
V |
19 | 17 | rabex 3892 |
. . . . . . . . . . . . . . . . 17
⊢ {u ∈
Q ∣ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q u} ∈
V |
20 | 18, 19 | op2nd 5716 |
. . . . . . . . . . . . . . . 16
⊢
(2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q (𝐹‘𝑞)}, {u
∈ Q ∣ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q u}〉) = {u
∈ Q ∣ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q u} |
21 | 16, 20 | eqtri 2057 |
. . . . . . . . . . . . . . 15
⊢
(2nd ‘𝐿) = {u
∈ Q ∣ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q u} |
22 | 15, 21 | elrab2 2694 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈
Q ∧ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
23 | 22 | biimpi 113 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ (2nd ‘𝐿) → (𝑠 ∈
Q ∧ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
24 | 23 | adantr 261 |
. . . . . . . . . . . 12
⊢ ((𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)) → (𝑠 ∈
Q ∧ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
25 | 24 | adantl 262 |
. . . . . . . . . . 11
⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) → (𝑠 ∈
Q ∧ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
26 | 25 | adantr 261 |
. . . . . . . . . 10
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 ∈
Q ∧ ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠)) |
27 | 26 | simpld 105 |
. . . . . . . . 9
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠 ∈
Q) |
28 | | vex 2554 |
. . . . . . . . . . . . . 14
⊢ 𝑡 ∈ V |
29 | | breq2 3759 |
. . . . . . . . . . . . . 14
⊢ (u = 𝑡 → (𝑆 <Q u ↔ 𝑆 <Q 𝑡)) |
30 | | ltnqex 6531 |
. . . . . . . . . . . . . . 15
⊢ {𝑙 ∣ 𝑙 <Q 𝑆} ∈ V |
31 | | gtnqex 6532 |
. . . . . . . . . . . . . . 15
⊢ {u ∣ 𝑆 <Q u} ∈
V |
32 | 30, 31 | op2nd 5716 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉) = {u
∣ 𝑆
<Q u} |
33 | 28, 29, 32 | elab2 2684 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉) ↔ 𝑆 <Q 𝑡) |
34 | | ltrelnq 6349 |
. . . . . . . . . . . . . 14
⊢
<Q ⊆ (Q ×
Q) |
35 | 34 | brel 4335 |
. . . . . . . . . . . . 13
⊢ (𝑆 <Q
𝑡 → (𝑆 ∈
Q ∧ 𝑡 ∈
Q)) |
36 | 33, 35 | sylbi 114 |
. . . . . . . . . . . 12
⊢ (𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉) → (𝑆 ∈
Q ∧ 𝑡 ∈
Q)) |
37 | 36 | simprd 107 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉) → 𝑡 ∈
Q) |
38 | 37 | ad2antll 460 |
. . . . . . . . . 10
⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) → 𝑡 ∈
Q) |
39 | 38 | adantr 261 |
. . . . . . . . 9
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 ∈
Q) |
40 | | addclnq 6359 |
. . . . . . . . 9
⊢ ((𝑠 ∈ Q ∧ 𝑡
∈ Q) → (𝑠 +Q 𝑡) ∈
Q) |
41 | 27, 39, 40 | syl2anc 391 |
. . . . . . . 8
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) ∈
Q) |
42 | | eleq1 2097 |
. . . . . . . . 9
⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 ∈
Q ↔ (𝑠
+Q 𝑡)
∈ Q)) |
43 | 42 | adantl 262 |
. . . . . . . 8
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟 ∈
Q ↔ (𝑠
+Q 𝑡)
∈ Q)) |
44 | 41, 43 | mpbird 156 |
. . . . . . 7
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈
Q) |
45 | 26 | simprd 107 |
. . . . . . . 8
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) |
46 | 33 | biimpi 113 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉) → 𝑆 <Q 𝑡) |
47 | 46 | ad2antll 460 |
. . . . . . . . . . . . . . 15
⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) → 𝑆 <Q 𝑡) |
48 | 47 | adantr 261 |
. . . . . . . . . . . . . 14
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑆 <Q 𝑡) |
49 | 48 | ad2antrr 457 |
. . . . . . . . . . . . 13
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑆 <Q 𝑡) |
50 | 6 | ad5antr 465 |
. . . . . . . . . . . . . 14
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑆 ∈
Q) |
51 | 39 | ad2antrr 457 |
. . . . . . . . . . . . . 14
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑡 ∈
Q) |
52 | 1 | ad5antr 465 |
. . . . . . . . . . . . . . . 16
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝐹:Q⟶Q) |
53 | | simplr 482 |
. . . . . . . . . . . . . . . 16
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑞 ∈
Q) |
54 | 52, 53 | ffvelrnd 5246 |
. . . . . . . . . . . . . . 15
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (𝐹‘𝑞) ∈
Q) |
55 | | addclnq 6359 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹‘𝑞) ∈
Q ∧ 𝑞 ∈
Q) → ((𝐹‘𝑞) +Q 𝑞) ∈
Q) |
56 | 54, 53, 55 | syl2anc 391 |
. . . . . . . . . . . . . 14
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹‘𝑞) +Q 𝑞) ∈
Q) |
57 | | ltanqg 6384 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ∈ Q ∧ 𝑡
∈ Q ∧ ((𝐹‘𝑞) +Q 𝑞) ∈
Q) → (𝑆
<Q 𝑡 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
(((𝐹‘𝑞) +Q 𝑞) +Q 𝑡))) |
58 | 50, 51, 56, 57 | syl3anc 1134 |
. . . . . . . . . . . . 13
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (𝑆 <Q 𝑡 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
(((𝐹‘𝑞) +Q 𝑞) +Q 𝑡))) |
59 | 49, 58 | mpbid 135 |
. . . . . . . . . . . 12
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
(((𝐹‘𝑞) +Q 𝑞) +Q 𝑡)) |
60 | | simpr 103 |
. . . . . . . . . . . . 13
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) |
61 | | ltanqg 6384 |
. . . . . . . . . . . . . . 15
⊢
((f ∈ Q ∧ g ∈ Q ∧ ℎ
∈ Q) → (f <Q g ↔ (ℎ +Q f) <Q (ℎ +Q g))) |
62 | 61 | adantl 262 |
. . . . . . . . . . . . . 14
⊢
(((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) ∧ (f ∈ Q ∧ g ∈ Q ∧ ℎ
∈ Q)) → (f <Q g ↔ (ℎ +Q f) <Q (ℎ +Q g))) |
63 | 27 | ad2antrr 457 |
. . . . . . . . . . . . . 14
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑠 ∈
Q) |
64 | | addcomnqg 6365 |
. . . . . . . . . . . . . . 15
⊢
((f ∈ Q ∧ g ∈ Q) → (f +Q g) = (g
+Q f)) |
65 | 64 | adantl 262 |
. . . . . . . . . . . . . 14
⊢
(((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) ∧ (f ∈ Q ∧ g ∈ Q)) → (f +Q g) = (g
+Q f)) |
66 | 62, 56, 63, 51, 65 | caovord2d 5612 |
. . . . . . . . . . . . 13
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡))) |
67 | 60, 66 | mpbid 135 |
. . . . . . . . . . . 12
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡)) |
68 | | ltsonq 6382 |
. . . . . . . . . . . . 13
⊢
<Q Or Q |
69 | 68, 34 | sotri 4663 |
. . . . . . . . . . . 12
⊢
(((((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
(((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) ∧ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑡) <Q (𝑠 +Q 𝑡)) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
(𝑠 +Q
𝑡)) |
70 | 59, 67, 69 | syl2anc 391 |
. . . . . . . . . . 11
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
(𝑠 +Q
𝑡)) |
71 | | simpllr 486 |
. . . . . . . . . . 11
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → 𝑟 = (𝑠 +Q 𝑡)) |
72 | 70, 71 | breqtrrd 3781 |
. . . . . . . . . 10
⊢
((((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) ∧ ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠) → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
𝑟) |
73 | 72 | ex 108 |
. . . . . . . . 9
⊢
(((((φ ∧ 𝑟
∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) ∧ 𝑞 ∈ Q) → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
𝑟)) |
74 | 73 | reximdva 2415 |
. . . . . . . 8
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (∃𝑞 ∈
Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑠 → ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
𝑟)) |
75 | 45, 74 | mpd 13 |
. . . . . . 7
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
𝑟) |
76 | | breq2 3759 |
. . . . . . . . 9
⊢ (u = 𝑟 → ((((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u ↔ (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
𝑟)) |
77 | 76 | rexbidv 2321 |
. . . . . . . 8
⊢ (u = 𝑟 → (∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u ↔ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
𝑟)) |
78 | 17 | rabex 3892 |
. . . . . . . . 9
⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)} ∈ V |
79 | 17 | rabex 3892 |
. . . . . . . . 9
⊢ {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u} ∈
V |
80 | 78, 79 | op2nd 5716 |
. . . . . . . 8
⊢
(2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉) = {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u} |
81 | 77, 80 | elrab2 2694 |
. . . . . . 7
⊢ (𝑟 ∈ (2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉) ↔ (𝑟 ∈
Q ∧ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
𝑟)) |
82 | 44, 75, 81 | sylanbrc 394 |
. . . . . 6
⊢ ((((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈
(2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉)) |
83 | 82 | ex 108 |
. . . . 5
⊢ (((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) ∧
(𝑠 ∈ (2nd ‘𝐿) ∧ 𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈
(2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉))) |
84 | 83 | rexlimdvva 2434 |
. . . 4
⊢ ((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) → (∃𝑠 ∈
(2nd ‘𝐿)∃𝑡 ∈ (2nd ‘〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈
(2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉))) |
85 | 13, 84 | mpd 13 |
. . 3
⊢ ((φ ∧ 𝑟 ∈ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉))) → 𝑟 ∈
(2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉)) |
86 | 85 | ex 108 |
. 2
⊢ (φ → (𝑟 ∈
(2nd ‘(𝐿
+P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {u ∣ 𝑆 <Q u}〉)) → 𝑟 ∈
(2nd ‘〈{𝑙 ∈
Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉))) |
87 | 86 | ssrdv 2945 |
1
⊢ (φ → (2nd ‘(𝐿 +P
〈{𝑙 ∣ 𝑙 <Q
𝑆}, {u ∣ 𝑆 <Q u}〉)) ⊆ (2nd
‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈
Q (𝑙
+Q 𝑞)
<Q ((𝐹‘𝑞) +Q 𝑆)}, {u ∈
Q ∣ ∃𝑞 ∈
Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q
u}〉)) |