Step |
Hyp |
Ref |
Expression |
1 |
|
erdsze.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
2 |
|
erdsze.f |
⊢ ( 𝜑 → 𝐹 : ( 1 ... 𝑁 ) –1-1→ ℝ ) |
3 |
|
erdszelem.k |
⊢ 𝐾 = ( 𝑥 ∈ ( 1 ... 𝑁 ) ↦ sup ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝑥 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝑥 ∈ 𝑦 ) } ) , ℝ , < ) ) |
4 |
|
erdszelem.o |
⊢ 𝑂 Or ℝ |
5 |
|
erdszelem.a |
⊢ ( 𝜑 → 𝐴 ∈ ( 1 ... 𝑁 ) ) |
6 |
|
erdszelem.b |
⊢ ( 𝜑 → 𝐵 ∈ ( 1 ... 𝑁 ) ) |
7 |
|
erdszelem.l |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
8 |
|
hashf |
⊢ ♯ : V ⟶ ( ℕ0 ∪ { +∞ } ) |
9 |
|
ffun |
⊢ ( ♯ : V ⟶ ( ℕ0 ∪ { +∞ } ) → Fun ♯ ) |
10 |
8 9
|
ax-mp |
⊢ Fun ♯ |
11 |
1 2 3 4
|
erdszelem5 |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 1 ... 𝑁 ) ) → ( 𝐾 ‘ 𝐴 ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ) ) |
12 |
5 11
|
mpdan |
⊢ ( 𝜑 → ( 𝐾 ‘ 𝐴 ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ) ) |
13 |
|
fvelima |
⊢ ( ( Fun ♯ ∧ ( 𝐾 ‘ 𝐴 ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ) ) → ∃ 𝑓 ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ( ♯ ‘ 𝑓 ) = ( 𝐾 ‘ 𝐴 ) ) |
14 |
10 12 13
|
sylancr |
⊢ ( 𝜑 → ∃ 𝑓 ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ( ♯ ‘ 𝑓 ) = ( 𝐾 ‘ 𝐴 ) ) |
15 |
|
eqid |
⊢ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } = { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } |
16 |
15
|
erdszelem1 |
⊢ ( 𝑓 ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ↔ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) |
17 |
|
fzfid |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 1 ... 𝐴 ) ∈ Fin ) |
18 |
|
simplr1 |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝑓 ⊆ ( 1 ... 𝐴 ) ) |
19 |
|
ssfi |
⊢ ( ( ( 1 ... 𝐴 ) ∈ Fin ∧ 𝑓 ⊆ ( 1 ... 𝐴 ) ) → 𝑓 ∈ Fin ) |
20 |
17 18 19
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝑓 ∈ Fin ) |
21 |
|
hashcl |
⊢ ( 𝑓 ∈ Fin → ( ♯ ‘ 𝑓 ) ∈ ℕ0 ) |
22 |
20 21
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ ‘ 𝑓 ) ∈ ℕ0 ) |
23 |
22
|
nn0red |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ ‘ 𝑓 ) ∈ ℝ ) |
24 |
|
eqid |
⊢ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } = { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } |
25 |
24
|
erdszelem2 |
⊢ ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ∈ Fin ∧ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ⊆ ℕ ) |
26 |
25
|
simpri |
⊢ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ⊆ ℕ |
27 |
|
nnssre |
⊢ ℕ ⊆ ℝ |
28 |
26 27
|
sstri |
⊢ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ⊆ ℝ |
29 |
28
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ⊆ ℝ ) |
30 |
5
|
elfzelzd |
⊢ ( 𝜑 → 𝐴 ∈ ℤ ) |
31 |
6
|
elfzelzd |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
32 |
|
elfznn |
⊢ ( 𝐴 ∈ ( 1 ... 𝑁 ) → 𝐴 ∈ ℕ ) |
33 |
5 32
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ ℕ ) |
34 |
33
|
nnred |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
35 |
|
elfznn |
⊢ ( 𝐵 ∈ ( 1 ... 𝑁 ) → 𝐵 ∈ ℕ ) |
36 |
6 35
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ ℕ ) |
37 |
36
|
nnred |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
38 |
34 37 7
|
ltled |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
39 |
|
eluz2 |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ↔ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵 ) ) |
40 |
30 31 38 39
|
syl3anbrc |
⊢ ( 𝜑 → 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) |
41 |
|
fzss2 |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 1 ... 𝐴 ) ⊆ ( 1 ... 𝐵 ) ) |
42 |
40 41
|
syl |
⊢ ( 𝜑 → ( 1 ... 𝐴 ) ⊆ ( 1 ... 𝐵 ) ) |
43 |
42
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 1 ... 𝐴 ) ⊆ ( 1 ... 𝐵 ) ) |
44 |
18 43
|
sstrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝑓 ⊆ ( 1 ... 𝐵 ) ) |
45 |
|
elfz1end |
⊢ ( 𝐵 ∈ ℕ ↔ 𝐵 ∈ ( 1 ... 𝐵 ) ) |
46 |
36 45
|
sylib |
⊢ ( 𝜑 → 𝐵 ∈ ( 1 ... 𝐵 ) ) |
47 |
46
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝐵 ∈ ( 1 ... 𝐵 ) ) |
48 |
47
|
snssd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → { 𝐵 } ⊆ ( 1 ... 𝐵 ) ) |
49 |
44 48
|
unssd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝑓 ∪ { 𝐵 } ) ⊆ ( 1 ... 𝐵 ) ) |
50 |
|
simplr2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ) |
51 |
|
f1f |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) –1-1→ ℝ → 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ) |
52 |
2 51
|
syl |
⊢ ( 𝜑 → 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ) |
54 |
|
elfzuz3 |
⊢ ( 𝐴 ∈ ( 1 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝐴 ) ) |
55 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 1 ... 𝐴 ) ⊆ ( 1 ... 𝑁 ) ) |
56 |
5 54 55
|
3syl |
⊢ ( 𝜑 → ( 1 ... 𝐴 ) ⊆ ( 1 ... 𝑁 ) ) |
57 |
56
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 1 ... 𝐴 ) ⊆ ( 1 ... 𝑁 ) ) |
58 |
18 57
|
sstrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝑓 ⊆ ( 1 ... 𝑁 ) ) |
59 |
|
fzssuz |
⊢ ( 1 ... 𝑁 ) ⊆ ( ℤ≥ ‘ 1 ) |
60 |
|
uzssz |
⊢ ( ℤ≥ ‘ 1 ) ⊆ ℤ |
61 |
|
zssre |
⊢ ℤ ⊆ ℝ |
62 |
60 61
|
sstri |
⊢ ( ℤ≥ ‘ 1 ) ⊆ ℝ |
63 |
59 62
|
sstri |
⊢ ( 1 ... 𝑁 ) ⊆ ℝ |
64 |
|
ltso |
⊢ < Or ℝ |
65 |
|
soss |
⊢ ( ( 1 ... 𝑁 ) ⊆ ℝ → ( < Or ℝ → < Or ( 1 ... 𝑁 ) ) ) |
66 |
63 64 65
|
mp2 |
⊢ < Or ( 1 ... 𝑁 ) |
67 |
|
soisores |
⊢ ( ( ( < Or ( 1 ... 𝑁 ) ∧ 𝑂 Or ℝ ) ∧ ( 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ∧ 𝑓 ⊆ ( 1 ... 𝑁 ) ) ) → ( ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ↔ ∀ 𝑧 ∈ 𝑓 ∀ 𝑤 ∈ 𝑓 ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
68 |
66 4 67
|
mpanl12 |
⊢ ( ( 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ∧ 𝑓 ⊆ ( 1 ... 𝑁 ) ) → ( ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ↔ ∀ 𝑧 ∈ 𝑓 ∀ 𝑤 ∈ 𝑓 ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
69 |
53 58 68
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ↔ ∀ 𝑧 ∈ 𝑓 ∀ 𝑤 ∈ 𝑓 ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
70 |
50 69
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ∀ 𝑧 ∈ 𝑓 ∀ 𝑤 ∈ 𝑓 ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
71 |
70
|
r19.21bi |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ∀ 𝑤 ∈ 𝑓 ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
72 |
18
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝑧 ∈ ( 1 ... 𝐴 ) ) |
73 |
|
elfzle2 |
⊢ ( 𝑧 ∈ ( 1 ... 𝐴 ) → 𝑧 ≤ 𝐴 ) |
74 |
72 73
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝑧 ≤ 𝐴 ) |
75 |
58
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝑧 ∈ ( 1 ... 𝑁 ) ) |
76 |
63 75
|
sseldi |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝑧 ∈ ℝ ) |
77 |
5
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝐴 ∈ ( 1 ... 𝑁 ) ) |
78 |
77 32
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝐴 ∈ ℕ ) |
79 |
78
|
nnred |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝐴 ∈ ℝ ) |
80 |
76 79
|
lenltd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝑧 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑧 ) ) |
81 |
74 80
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ¬ 𝐴 < 𝑧 ) |
82 |
50
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ) |
83 |
|
simplr3 |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝐴 ∈ 𝑓 ) |
84 |
83
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝐴 ∈ 𝑓 ) |
85 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝑧 ∈ 𝑓 ) |
86 |
|
isorel |
⊢ ( ( ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ ( 𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓 ) ) → ( 𝐴 < 𝑧 ↔ ( ( 𝐹 ↾ 𝑓 ) ‘ 𝐴 ) 𝑂 ( ( 𝐹 ↾ 𝑓 ) ‘ 𝑧 ) ) ) |
87 |
|
fvres |
⊢ ( 𝐴 ∈ 𝑓 → ( ( 𝐹 ↾ 𝑓 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) |
88 |
|
fvres |
⊢ ( 𝑧 ∈ 𝑓 → ( ( 𝐹 ↾ 𝑓 ) ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
89 |
87 88
|
breqan12d |
⊢ ( ( 𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓 ) → ( ( ( 𝐹 ↾ 𝑓 ) ‘ 𝐴 ) 𝑂 ( ( 𝐹 ↾ 𝑓 ) ‘ 𝑧 ) ↔ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ) ) |
90 |
89
|
adantl |
⊢ ( ( ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ ( 𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓 ) ) → ( ( ( 𝐹 ↾ 𝑓 ) ‘ 𝐴 ) 𝑂 ( ( 𝐹 ↾ 𝑓 ) ‘ 𝑧 ) ↔ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ) ) |
91 |
86 90
|
bitrd |
⊢ ( ( ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ ( 𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓 ) ) → ( 𝐴 < 𝑧 ↔ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ) ) |
92 |
82 84 85 91
|
syl12anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝐴 < 𝑧 ↔ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ) ) |
93 |
81 92
|
mtbid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ¬ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ) |
94 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) |
95 |
53
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ) |
96 |
95 75
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝐹 ‘ 𝑧 ) ∈ ℝ ) |
97 |
95 77
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ℝ ) |
98 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝐵 ∈ ( 1 ... 𝑁 ) ) |
99 |
98
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → 𝐵 ∈ ( 1 ... 𝑁 ) ) |
100 |
95 99
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝐹 ‘ 𝐵 ) ∈ ℝ ) |
101 |
|
sotr2 |
⊢ ( ( 𝑂 Or ℝ ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝐴 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝐵 ) ∈ ℝ ) ) → ( ( ¬ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ) |
102 |
4 101
|
mpan |
⊢ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝐴 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝐵 ) ∈ ℝ ) → ( ( ¬ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ) |
103 |
96 97 100 102
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( ( ¬ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ) |
104 |
93 94 103
|
mp2and |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) |
105 |
104
|
a1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ) |
106 |
|
elsni |
⊢ ( 𝑤 ∈ { 𝐵 } → 𝑤 = 𝐵 ) |
107 |
106
|
fveq2d |
⊢ ( 𝑤 ∈ { 𝐵 } → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝐵 ) ) |
108 |
107
|
breq2d |
⊢ ( 𝑤 ∈ { 𝐵 } → ( ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ↔ ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ) |
109 |
108
|
imbi2d |
⊢ ( 𝑤 ∈ { 𝐵 } → ( ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ↔ ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ) ) |
110 |
105 109
|
syl5ibrcom |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ( 𝑤 ∈ { 𝐵 } → ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
111 |
110
|
ralrimiv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ∀ 𝑤 ∈ { 𝐵 } ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
112 |
|
ralunb |
⊢ ( ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ↔ ( ∀ 𝑤 ∈ 𝑓 ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ∧ ∀ 𝑤 ∈ { 𝐵 } ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
113 |
71 111 112
|
sylanbrc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑧 ∈ 𝑓 ) → ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
114 |
113
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ∀ 𝑧 ∈ 𝑓 ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
115 |
49
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ) → 𝑤 ∈ ( 1 ... 𝐵 ) ) |
116 |
|
elfzle2 |
⊢ ( 𝑤 ∈ ( 1 ... 𝐵 ) → 𝑤 ≤ 𝐵 ) |
117 |
116
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 1 ... 𝐵 ) ) → 𝑤 ≤ 𝐵 ) |
118 |
|
elfzelz |
⊢ ( 𝑤 ∈ ( 1 ... 𝐵 ) → 𝑤 ∈ ℤ ) |
119 |
118
|
zred |
⊢ ( 𝑤 ∈ ( 1 ... 𝐵 ) → 𝑤 ∈ ℝ ) |
120 |
119
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 1 ... 𝐵 ) ) → 𝑤 ∈ ℝ ) |
121 |
37
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 1 ... 𝐵 ) ) → 𝐵 ∈ ℝ ) |
122 |
120 121
|
lenltd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 1 ... 𝐵 ) ) → ( 𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤 ) ) |
123 |
117 122
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 1 ... 𝐵 ) ) → ¬ 𝐵 < 𝑤 ) |
124 |
115 123
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ) → ¬ 𝐵 < 𝑤 ) |
125 |
124
|
pm2.21d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ∧ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ) → ( 𝐵 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
126 |
125
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝐵 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
127 |
|
elsni |
⊢ ( 𝑧 ∈ { 𝐵 } → 𝑧 = 𝐵 ) |
128 |
127
|
breq1d |
⊢ ( 𝑧 ∈ { 𝐵 } → ( 𝑧 < 𝑤 ↔ 𝐵 < 𝑤 ) ) |
129 |
128
|
imbi1d |
⊢ ( 𝑧 ∈ { 𝐵 } → ( ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ↔ ( 𝐵 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
130 |
129
|
ralbidv |
⊢ ( 𝑧 ∈ { 𝐵 } → ( ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ↔ ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝐵 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
131 |
126 130
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝑧 ∈ { 𝐵 } → ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
132 |
131
|
ralrimiv |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ∀ 𝑧 ∈ { 𝐵 } ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
133 |
|
ralunb |
⊢ ( ∀ 𝑧 ∈ ( 𝑓 ∪ { 𝐵 } ) ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ↔ ( ∀ 𝑧 ∈ 𝑓 ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ∧ ∀ 𝑧 ∈ { 𝐵 } ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
134 |
114 132 133
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ∀ 𝑧 ∈ ( 𝑓 ∪ { 𝐵 } ) ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) |
135 |
98
|
snssd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → { 𝐵 } ⊆ ( 1 ... 𝑁 ) ) |
136 |
58 135
|
unssd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝑓 ∪ { 𝐵 } ) ⊆ ( 1 ... 𝑁 ) ) |
137 |
|
soisores |
⊢ ( ( ( < Or ( 1 ... 𝑁 ) ∧ 𝑂 Or ℝ ) ∧ ( 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ∧ ( 𝑓 ∪ { 𝐵 } ) ⊆ ( 1 ... 𝑁 ) ) ) → ( ( 𝐹 ↾ ( 𝑓 ∪ { 𝐵 } ) ) Isom < , 𝑂 ( ( 𝑓 ∪ { 𝐵 } ) , ( 𝐹 “ ( 𝑓 ∪ { 𝐵 } ) ) ) ↔ ∀ 𝑧 ∈ ( 𝑓 ∪ { 𝐵 } ) ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
138 |
66 4 137
|
mpanl12 |
⊢ ( ( 𝐹 : ( 1 ... 𝑁 ) ⟶ ℝ ∧ ( 𝑓 ∪ { 𝐵 } ) ⊆ ( 1 ... 𝑁 ) ) → ( ( 𝐹 ↾ ( 𝑓 ∪ { 𝐵 } ) ) Isom < , 𝑂 ( ( 𝑓 ∪ { 𝐵 } ) , ( 𝐹 “ ( 𝑓 ∪ { 𝐵 } ) ) ) ↔ ∀ 𝑧 ∈ ( 𝑓 ∪ { 𝐵 } ) ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
139 |
53 136 138
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ( 𝐹 ↾ ( 𝑓 ∪ { 𝐵 } ) ) Isom < , 𝑂 ( ( 𝑓 ∪ { 𝐵 } ) , ( 𝐹 “ ( 𝑓 ∪ { 𝐵 } ) ) ) ↔ ∀ 𝑧 ∈ ( 𝑓 ∪ { 𝐵 } ) ∀ 𝑤 ∈ ( 𝑓 ∪ { 𝐵 } ) ( 𝑧 < 𝑤 → ( 𝐹 ‘ 𝑧 ) 𝑂 ( 𝐹 ‘ 𝑤 ) ) ) ) |
140 |
134 139
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐹 ↾ ( 𝑓 ∪ { 𝐵 } ) ) Isom < , 𝑂 ( ( 𝑓 ∪ { 𝐵 } ) , ( 𝐹 “ ( 𝑓 ∪ { 𝐵 } ) ) ) ) |
141 |
|
ssun2 |
⊢ { 𝐵 } ⊆ ( 𝑓 ∪ { 𝐵 } ) |
142 |
|
snssg |
⊢ ( 𝐵 ∈ ( 1 ... 𝐵 ) → ( 𝐵 ∈ ( 𝑓 ∪ { 𝐵 } ) ↔ { 𝐵 } ⊆ ( 𝑓 ∪ { 𝐵 } ) ) ) |
143 |
47 142
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐵 ∈ ( 𝑓 ∪ { 𝐵 } ) ↔ { 𝐵 } ⊆ ( 𝑓 ∪ { 𝐵 } ) ) ) |
144 |
141 143
|
mpbiri |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → 𝐵 ∈ ( 𝑓 ∪ { 𝐵 } ) ) |
145 |
24
|
erdszelem1 |
⊢ ( ( 𝑓 ∪ { 𝐵 } ) ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ↔ ( ( 𝑓 ∪ { 𝐵 } ) ⊆ ( 1 ... 𝐵 ) ∧ ( 𝐹 ↾ ( 𝑓 ∪ { 𝐵 } ) ) Isom < , 𝑂 ( ( 𝑓 ∪ { 𝐵 } ) , ( 𝐹 “ ( 𝑓 ∪ { 𝐵 } ) ) ) ∧ 𝐵 ∈ ( 𝑓 ∪ { 𝐵 } ) ) ) |
146 |
49 140 144 145
|
syl3anbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝑓 ∪ { 𝐵 } ) ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) |
147 |
|
vex |
⊢ 𝑓 ∈ V |
148 |
|
snex |
⊢ { 𝐵 } ∈ V |
149 |
147 148
|
unex |
⊢ ( 𝑓 ∪ { 𝐵 } ) ∈ V |
150 |
8
|
fdmi |
⊢ dom ♯ = V |
151 |
149 150
|
eleqtrri |
⊢ ( 𝑓 ∪ { 𝐵 } ) ∈ dom ♯ |
152 |
|
funfvima |
⊢ ( ( Fun ♯ ∧ ( 𝑓 ∪ { 𝐵 } ) ∈ dom ♯ ) → ( ( 𝑓 ∪ { 𝐵 } ) ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } → ( ♯ ‘ ( 𝑓 ∪ { 𝐵 } ) ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ) ) |
153 |
10 151 152
|
mp2an |
⊢ ( ( 𝑓 ∪ { 𝐵 } ) ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } → ( ♯ ‘ ( 𝑓 ∪ { 𝐵 } ) ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ) |
154 |
146 153
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ ‘ ( 𝑓 ∪ { 𝐵 } ) ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ) |
155 |
154
|
ne0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ≠ ∅ ) |
156 |
25
|
simpli |
⊢ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ∈ Fin |
157 |
|
fimaxre2 |
⊢ ( ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ⊆ ℝ ∧ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ∈ Fin ) → ∃ 𝑧 ∈ ℝ ∀ 𝑤 ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) 𝑤 ≤ 𝑧 ) |
158 |
29 156 157
|
sylancl |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ∃ 𝑧 ∈ ℝ ∀ 𝑤 ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) 𝑤 ≤ 𝑧 ) |
159 |
34 37
|
ltnled |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) ) |
160 |
7 159
|
mpbid |
⊢ ( 𝜑 → ¬ 𝐵 ≤ 𝐴 ) |
161 |
|
elfzle2 |
⊢ ( 𝐵 ∈ ( 1 ... 𝐴 ) → 𝐵 ≤ 𝐴 ) |
162 |
160 161
|
nsyl |
⊢ ( 𝜑 → ¬ 𝐵 ∈ ( 1 ... 𝐴 ) ) |
163 |
162
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ¬ 𝐵 ∈ ( 1 ... 𝐴 ) ) |
164 |
18 163
|
ssneldd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ¬ 𝐵 ∈ 𝑓 ) |
165 |
|
hashunsng |
⊢ ( 𝐵 ∈ ( 1 ... 𝑁 ) → ( ( 𝑓 ∈ Fin ∧ ¬ 𝐵 ∈ 𝑓 ) → ( ♯ ‘ ( 𝑓 ∪ { 𝐵 } ) ) = ( ( ♯ ‘ 𝑓 ) + 1 ) ) ) |
166 |
98 165
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ( 𝑓 ∈ Fin ∧ ¬ 𝐵 ∈ 𝑓 ) → ( ♯ ‘ ( 𝑓 ∪ { 𝐵 } ) ) = ( ( ♯ ‘ 𝑓 ) + 1 ) ) ) |
167 |
20 164 166
|
mp2and |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ ‘ ( 𝑓 ∪ { 𝐵 } ) ) = ( ( ♯ ‘ 𝑓 ) + 1 ) ) |
168 |
167 154
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ( ♯ ‘ 𝑓 ) + 1 ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ) |
169 |
|
suprub |
⊢ ( ( ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ⊆ ℝ ∧ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ≠ ∅ ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑤 ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) 𝑤 ≤ 𝑧 ) ∧ ( ( ♯ ‘ 𝑓 ) + 1 ) ∈ ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) ) → ( ( ♯ ‘ 𝑓 ) + 1 ) ≤ sup ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) , ℝ , < ) ) |
170 |
29 155 158 168 169
|
syl31anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ( ♯ ‘ 𝑓 ) + 1 ) ≤ sup ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) , ℝ , < ) ) |
171 |
1 2 3
|
erdszelem3 |
⊢ ( 𝐵 ∈ ( 1 ... 𝑁 ) → ( 𝐾 ‘ 𝐵 ) = sup ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) , ℝ , < ) ) |
172 |
6 171
|
syl |
⊢ ( 𝜑 → ( 𝐾 ‘ 𝐵 ) = sup ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) , ℝ , < ) ) |
173 |
172
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐾 ‘ 𝐵 ) = sup ( ( ♯ “ { 𝑦 ∈ 𝒫 ( 1 ... 𝐵 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐵 ∈ 𝑦 ) } ) , ℝ , < ) ) |
174 |
170 173
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ( ♯ ‘ 𝑓 ) + 1 ) ≤ ( 𝐾 ‘ 𝐵 ) ) |
175 |
1 2 3 4
|
erdszelem6 |
⊢ ( 𝜑 → 𝐾 : ( 1 ... 𝑁 ) ⟶ ℕ ) |
176 |
175 6
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐾 ‘ 𝐵 ) ∈ ℕ ) |
177 |
176
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐾 ‘ 𝐵 ) ∈ ℕ ) |
178 |
177
|
nnnn0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( 𝐾 ‘ 𝐵 ) ∈ ℕ0 ) |
179 |
|
nn0ltp1le |
⊢ ( ( ( ♯ ‘ 𝑓 ) ∈ ℕ0 ∧ ( 𝐾 ‘ 𝐵 ) ∈ ℕ0 ) → ( ( ♯ ‘ 𝑓 ) < ( 𝐾 ‘ 𝐵 ) ↔ ( ( ♯ ‘ 𝑓 ) + 1 ) ≤ ( 𝐾 ‘ 𝐵 ) ) ) |
180 |
22 178 179
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ( ♯ ‘ 𝑓 ) < ( 𝐾 ‘ 𝐵 ) ↔ ( ( ♯ ‘ 𝑓 ) + 1 ) ≤ ( 𝐾 ‘ 𝐵 ) ) ) |
181 |
174 180
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ ‘ 𝑓 ) < ( 𝐾 ‘ 𝐵 ) ) |
182 |
23 181
|
ltned |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) ∧ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) → ( ♯ ‘ 𝑓 ) ≠ ( 𝐾 ‘ 𝐵 ) ) |
183 |
182
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) → ( ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) → ( ♯ ‘ 𝑓 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ) |
184 |
|
neeq1 |
⊢ ( ( ♯ ‘ 𝑓 ) = ( 𝐾 ‘ 𝐴 ) → ( ( ♯ ‘ 𝑓 ) ≠ ( 𝐾 ‘ 𝐵 ) ↔ ( 𝐾 ‘ 𝐴 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ) |
185 |
184
|
imbi2d |
⊢ ( ( ♯ ‘ 𝑓 ) = ( 𝐾 ‘ 𝐴 ) → ( ( ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) → ( ♯ ‘ 𝑓 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ↔ ( ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) → ( 𝐾 ‘ 𝐴 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ) ) |
186 |
183 185
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( 𝑓 ⊆ ( 1 ... 𝐴 ) ∧ ( 𝐹 ↾ 𝑓 ) Isom < , 𝑂 ( 𝑓 , ( 𝐹 “ 𝑓 ) ) ∧ 𝐴 ∈ 𝑓 ) ) → ( ( ♯ ‘ 𝑓 ) = ( 𝐾 ‘ 𝐴 ) → ( ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) → ( 𝐾 ‘ 𝐴 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ) ) |
187 |
16 186
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ) → ( ( ♯ ‘ 𝑓 ) = ( 𝐾 ‘ 𝐴 ) → ( ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) → ( 𝐾 ‘ 𝐴 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ) ) |
188 |
187
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ { 𝑦 ∈ 𝒫 ( 1 ... 𝐴 ) ∣ ( ( 𝐹 ↾ 𝑦 ) Isom < , 𝑂 ( 𝑦 , ( 𝐹 “ 𝑦 ) ) ∧ 𝐴 ∈ 𝑦 ) } ( ♯ ‘ 𝑓 ) = ( 𝐾 ‘ 𝐴 ) → ( ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) → ( 𝐾 ‘ 𝐴 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ) ) |
189 |
14 188
|
mpd |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) → ( 𝐾 ‘ 𝐴 ) ≠ ( 𝐾 ‘ 𝐵 ) ) ) |
190 |
189
|
necon2bd |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐴 ) = ( 𝐾 ‘ 𝐵 ) → ¬ ( 𝐹 ‘ 𝐴 ) 𝑂 ( 𝐹 ‘ 𝐵 ) ) ) |