Step | Hyp | Ref
| Expression |
1 | | erdsze2lem.n |
. . . . 5
⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1)) |
2 | | erdsze2.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ℕ) |
3 | | nnm1nn0 11334 |
. . . . . . 7
⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈
ℕ0) |
4 | 2, 3 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑅 − 1) ∈
ℕ0) |
5 | | erdsze2.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ℕ) |
6 | | nnm1nn0 11334 |
. . . . . . 7
⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈
ℕ0) |
7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑆 − 1) ∈
ℕ0) |
8 | 4, 7 | nn0mulcld 11356 |
. . . . 5
⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈
ℕ0) |
9 | 1, 8 | syl5eqel 2705 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
10 | | nn0p1nn 11332 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ) |
11 | 9, 10 | syl 17 |
. . 3
⊢ (𝜑 → (𝑁 + 1) ∈ ℕ) |
12 | | erdsze2.f |
. . . 4
⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) |
13 | | erdsze2lem.g |
. . . 4
⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))–1-1→𝐴) |
14 | | f1co 6110 |
. . . 4
⊢ ((𝐹:𝐴–1-1→ℝ ∧ 𝐺:(1...(𝑁 + 1))–1-1→𝐴) → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))–1-1→ℝ) |
15 | 12, 13, 14 | syl2anc 693 |
. . 3
⊢ (𝜑 → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))–1-1→ℝ) |
16 | 9 | nn0red 11352 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℝ) |
17 | 16 | ltp1d 10954 |
. . . 4
⊢ (𝜑 → 𝑁 < (𝑁 + 1)) |
18 | 1, 17 | syl5eqbrr 4689 |
. . 3
⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (𝑁 + 1)) |
19 | 11, 15, 2, 5, 18 | erdsze 31184 |
. 2
⊢ (𝜑 → ∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))))) |
20 | | selpw 4165 |
. . . 4
⊢ (𝑡 ∈ 𝒫 (1...(𝑁 + 1)) ↔ 𝑡 ⊆ (1...(𝑁 + 1))) |
21 | | imassrn 5477 |
. . . . . . . 8
⊢ (𝐺 “ 𝑡) ⊆ ran 𝐺 |
22 | | f1f 6101 |
. . . . . . . . . 10
⊢ (𝐺:(1...(𝑁 + 1))–1-1→𝐴 → 𝐺:(1...(𝑁 + 1))⟶𝐴) |
23 | 13, 22 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))⟶𝐴) |
24 | | frn 6053 |
. . . . . . . . 9
⊢ (𝐺:(1...(𝑁 + 1))⟶𝐴 → ran 𝐺 ⊆ 𝐴) |
25 | 23, 24 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ran 𝐺 ⊆ 𝐴) |
26 | 21, 25 | syl5ss 3614 |
. . . . . . 7
⊢ (𝜑 → (𝐺 “ 𝑡) ⊆ 𝐴) |
27 | | erdsze2.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
28 | | reex 10027 |
. . . . . . . . 9
⊢ ℝ
∈ V |
29 | | ssexg 4804 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧ ℝ
∈ V) → 𝐴 ∈
V) |
30 | 27, 28, 29 | sylancl 694 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ V) |
31 | | elpw2g 4827 |
. . . . . . . 8
⊢ (𝐴 ∈ V → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴)) |
32 | 30, 31 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴)) |
33 | 26, 32 | mpbird 247 |
. . . . . 6
⊢ (𝜑 → (𝐺 “ 𝑡) ∈ 𝒫 𝐴) |
34 | 33 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ 𝒫 𝐴) |
35 | | vex 3203 |
. . . . . . . . . . . 12
⊢ 𝑡 ∈ V |
36 | 35 | f1imaen 8018 |
. . . . . . . . . . 11
⊢ ((𝐺:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡) |
37 | 13, 36 | sylan 488 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡) |
38 | | fzfid 12772 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ∈ Fin) |
39 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ⊆ (1...(𝑁 + 1))) |
40 | | ssfi 8180 |
. . . . . . . . . . . . 13
⊢
(((1...(𝑁 + 1))
∈ Fin ∧ 𝑡 ⊆
(1...(𝑁 + 1))) → 𝑡 ∈ Fin) |
41 | 38, 39, 40 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ∈ Fin) |
42 | | enfii 8177 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ Fin ∧ (𝐺 “ 𝑡) ≈ 𝑡) → (𝐺 “ 𝑡) ∈ Fin) |
43 | 41, 37, 42 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ Fin) |
44 | | hashen 13135 |
. . . . . . . . . . 11
⊢ (((𝐺 “ 𝑡) ∈ Fin ∧ 𝑡 ∈ Fin) → ((#‘(𝐺 “ 𝑡)) = (#‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡)) |
45 | 43, 41, 44 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((#‘(𝐺 “ 𝑡)) = (#‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡)) |
46 | 37, 45 | mpbird 247 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (#‘(𝐺 “ 𝑡)) = (#‘𝑡)) |
47 | 46 | breq2d 4665 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (#‘(𝐺 “ 𝑡)) ↔ 𝑅 ≤ (#‘𝑡))) |
48 | 47 | biimprd 238 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (#‘𝑡) → 𝑅 ≤ (#‘(𝐺 “ 𝑡)))) |
49 | | erdsze2lem.i |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺)) |
50 | 49 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺)) |
51 | 39 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (1...(𝑁 + 1))) |
52 | | simprl 794 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑡) |
53 | 51, 52 | sseldd 3604 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (1...(𝑁 + 1))) |
54 | | simprr 796 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡) |
55 | 51, 54 | sseldd 3604 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (1...(𝑁 + 1))) |
56 | | isorel 6576 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺) ∧ (𝑥 ∈ (1...(𝑁 + 1)) ∧ 𝑦 ∈ (1...(𝑁 + 1)))) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦))) |
57 | 50, 53, 55, 56 | syl12anc 1324 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦))) |
58 | 57 | biimpd 219 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
59 | 58 | ralrimivva 2971 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
60 | | elfznn 12370 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℕ) |
61 | 60 | nnred 11035 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℝ) |
62 | 61 | ssriv 3607 |
. . . . . . . . . . . . . 14
⊢
(1...(𝑁 + 1))
⊆ ℝ |
63 | 62 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ⊆ ℝ) |
64 | | ltso 10118 |
. . . . . . . . . . . . 13
⊢ < Or
ℝ |
65 | | soss 5053 |
. . . . . . . . . . . . 13
⊢
((1...(𝑁 + 1))
⊆ ℝ → ( < Or ℝ → < Or (1...(𝑁 + 1)))) |
66 | 63, 64, 65 | mpisyl 21 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or (1...(𝑁 + 1))) |
67 | 27 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐴 ⊆ ℝ) |
68 | | soss 5053 |
. . . . . . . . . . . . 13
⊢ (𝐴 ⊆ ℝ → ( <
Or ℝ → < Or 𝐴)) |
69 | 67, 64, 68 | mpisyl 21 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or 𝐴) |
70 | 23 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐺:(1...(𝑁 + 1))⟶𝐴) |
71 | | soisores 6577 |
. . . . . . . . . . . 12
⊢ ((( <
Or (1...(𝑁 + 1)) ∧ <
Or 𝐴) ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1)))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
72 | 66, 69, 70, 39, 71 | syl22anc 1327 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
73 | 59, 72 | mpbird 247 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡))) |
74 | | isocnv 6580 |
. . . . . . . . . 10
⊢ ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) → ◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡)) |
75 | 73, 74 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡)) |
76 | | isotr 6586 |
. . . . . . . . . 10
⊢ ((◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))) |
77 | 76 | ex 450 |
. . . . . . . . 9
⊢ (◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
78 | 75, 77 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
79 | | resco 5639 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∘ 𝐺) ↾ 𝑡) = (𝐹 ∘ (𝐺 ↾ 𝑡)) |
80 | 79 | coeq1i 5281 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ◡(𝐺 ↾ 𝑡)) |
81 | | coass 5654 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) |
82 | 80, 81 | eqtri 2644 |
. . . . . . . . . . 11
⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) |
83 | | f1ores 6151 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡)) |
84 | 13, 83 | sylan 488 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡)) |
85 | | f1ococnv2 6163 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡) → ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡))) |
86 | 84, 85 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡))) |
87 | 86 | coeq2d 5284 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) = (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡)))) |
88 | | coires1 5653 |
. . . . . . . . . . . 12
⊢ (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡)) |
89 | 87, 88 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡))) |
90 | 82, 89 | syl5eq 2668 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡))) |
91 | | isoeq1 6567 |
. . . . . . . . . 10
⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
92 | 90, 91 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
93 | | imaco 5640 |
. . . . . . . . . 10
⊢ ((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) |
94 | | isoeq5 6571 |
. . . . . . . . . 10
⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
95 | 93, 94 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) |
96 | 92, 95 | syl6bb 276 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
97 | 78, 96 | sylibd 229 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
98 | 48, 97 | anim12d 586 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑅 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑅 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
99 | 46 | breq2d 4665 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (#‘(𝐺 “ 𝑡)) ↔ 𝑆 ≤ (#‘𝑡))) |
100 | 99 | biimprd 238 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (#‘𝑡) → 𝑆 ≤ (#‘(𝐺 “ 𝑡)))) |
101 | | isotr 6586 |
. . . . . . . . . 10
⊢ ((◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))) |
102 | 101 | ex 450 |
. . . . . . . . 9
⊢ (◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
103 | 75, 102 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
104 | | isoeq1 6567 |
. . . . . . . . . 10
⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
105 | 90, 104 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
106 | | isoeq5 6571 |
. . . . . . . . . 10
⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
107 | 93, 106 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) |
108 | 105, 107 | syl6bb 276 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
109 | 103, 108 | sylibd 229 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
110 | 100, 109 | anim12d 586 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑆 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑆 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
111 | 98, 110 | orim12d 883 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ((𝑅 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))) |
112 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → (#‘𝑠) = (#‘(𝐺 “ 𝑡))) |
113 | 112 | breq2d 4665 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑅 ≤ (#‘𝑠) ↔ 𝑅 ≤ (#‘(𝐺 “ 𝑡)))) |
114 | | reseq2 5391 |
. . . . . . . . . 10
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡))) |
115 | | isoeq1 6567 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)))) |
116 | 114, 115 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)))) |
117 | | isoeq4 6570 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)))) |
118 | | imaeq2 5462 |
. . . . . . . . . 10
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡))) |
119 | | isoeq5 6571 |
. . . . . . . . . 10
⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
120 | 118, 119 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
121 | 116, 117,
120 | 3bitrd 294 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
122 | 113, 121 | anbi12d 747 |
. . . . . . 7
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑅 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑅 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
123 | 112 | breq2d 4665 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑆 ≤ (#‘𝑠) ↔ 𝑆 ≤ (#‘(𝐺 “ 𝑡)))) |
124 | | isoeq1 6567 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))) |
125 | 114, 124 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))) |
126 | | isoeq4 6570 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)))) |
127 | | isoeq5 6571 |
. . . . . . . . . 10
⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
128 | 118, 127 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
129 | 125, 126,
128 | 3bitrd 294 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
130 | 123, 129 | anbi12d 747 |
. . . . . . 7
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑆 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑆 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
131 | 122, 130 | orbi12d 746 |
. . . . . 6
⊢ (𝑠 = (𝐺 “ 𝑡) → (((𝑅 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))) ↔ ((𝑅 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))) |
132 | 131 | rspcev 3309 |
. . . . 5
⊢ (((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ∧ ((𝑅 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (#‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) |
133 | 34, 111, 132 | syl6an 568 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))) |
134 | 20, 133 | sylan2b 492 |
. . 3
⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 (1...(𝑁 + 1))) → (((𝑅 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))) |
135 | 134 | rexlimdva 3031 |
. 2
⊢ (𝜑 → (∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (#‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))) |
136 | 19, 135 | mpd 15 |
1
⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) |