Step |
Hyp |
Ref |
Expression |
1 |
|
gsumbagdiag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
gsumbagdiag.s |
⊢ 𝑆 = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } |
3 |
|
gsumbagdiag.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
4 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) |
5 |
|
breq1 |
⊢ ( 𝑥 = 𝑌 → ( 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ↔ 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) ) |
6 |
5
|
elrab |
⊢ ( 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ↔ ( 𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) ) |
7 |
4 6
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) ) |
8 |
7
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∈ 𝐷 ) |
9 |
7
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) |
10 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐹 ∈ 𝐷 ) |
11 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∈ 𝑆 ) |
12 |
|
breq1 |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 ∘r ≤ 𝐹 ↔ 𝑋 ∘r ≤ 𝐹 ) ) |
13 |
12 2
|
elrab2 |
⊢ ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹 ) ) |
14 |
11 13
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹 ) ) |
15 |
14
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∈ 𝐷 ) |
16 |
1
|
psrbagf |
⊢ ( 𝑋 ∈ 𝐷 → 𝑋 : 𝐼 ⟶ ℕ0 ) |
17 |
15 16
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 : 𝐼 ⟶ ℕ0 ) |
18 |
14
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∘r ≤ 𝐹 ) |
19 |
1
|
psrbagcon |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 : 𝐼 ⟶ ℕ0 ∧ 𝑋 ∘r ≤ 𝐹 ) → ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
20 |
10 17 18 19
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
21 |
20
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) |
22 |
1
|
psrbagf |
⊢ ( 𝐹 ∈ 𝐷 → 𝐹 : 𝐼 ⟶ ℕ0 ) |
23 |
10 22
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
24 |
23
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐹 Fn 𝐼 ) |
25 |
10 24
|
fndmexd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐼 ∈ V ) |
26 |
1
|
psrbagf |
⊢ ( 𝑌 ∈ 𝐷 → 𝑌 : 𝐼 ⟶ ℕ0 ) |
27 |
8 26
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 : 𝐼 ⟶ ℕ0 ) |
28 |
20
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ) |
29 |
1
|
psrbagf |
⊢ ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 → ( 𝐹 ∘f − 𝑋 ) : 𝐼 ⟶ ℕ0 ) |
30 |
28 29
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) : 𝐼 ⟶ ℕ0 ) |
31 |
|
nn0re |
⊢ ( 𝑢 ∈ ℕ0 → 𝑢 ∈ ℝ ) |
32 |
|
nn0re |
⊢ ( 𝑣 ∈ ℕ0 → 𝑣 ∈ ℝ ) |
33 |
|
nn0re |
⊢ ( 𝑤 ∈ ℕ0 → 𝑤 ∈ ℝ ) |
34 |
|
letr |
⊢ ( ( 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ ) → ( ( 𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤 ) → 𝑢 ≤ 𝑤 ) ) |
35 |
31 32 33 34
|
syl3an |
⊢ ( ( 𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0 ∧ 𝑤 ∈ ℕ0 ) → ( ( 𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤 ) → 𝑢 ≤ 𝑤 ) ) |
36 |
35
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ ( 𝑢 ∈ ℕ0 ∧ 𝑣 ∈ ℕ0 ∧ 𝑤 ∈ ℕ0 ) ) → ( ( 𝑢 ≤ 𝑣 ∧ 𝑣 ≤ 𝑤 ) → 𝑢 ≤ 𝑤 ) ) |
37 |
25 27 30 23 36
|
caoftrn |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( ( 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) → 𝑌 ∘r ≤ 𝐹 ) ) |
38 |
9 21 37
|
mp2and |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∘r ≤ 𝐹 ) |
39 |
|
breq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ∘r ≤ 𝐹 ↔ 𝑌 ∘r ≤ 𝐹 ) ) |
40 |
39 2
|
elrab2 |
⊢ ( 𝑌 ∈ 𝑆 ↔ ( 𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ 𝐹 ) ) |
41 |
8 38 40
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∈ 𝑆 ) |
42 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) ↔ 𝑋 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) ) ) |
43 |
17
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 ) |
44 |
27
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 ) |
45 |
23
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑧 ) ∈ ℕ0 ) |
46 |
|
nn0re |
⊢ ( ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 → ( 𝑋 ‘ 𝑧 ) ∈ ℝ ) |
47 |
|
nn0re |
⊢ ( ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 → ( 𝑌 ‘ 𝑧 ) ∈ ℝ ) |
48 |
|
nn0re |
⊢ ( ( 𝐹 ‘ 𝑧 ) ∈ ℕ0 → ( 𝐹 ‘ 𝑧 ) ∈ ℝ ) |
49 |
|
leaddsub2 |
⊢ ( ( ( 𝑋 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝑌 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) ∈ ℝ ) → ( ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ≤ ( 𝐹 ‘ 𝑧 ) ↔ ( 𝑌 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) ) |
50 |
|
leaddsub |
⊢ ( ( ( 𝑋 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝑌 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) ∈ ℝ ) → ( ( ( 𝑋 ‘ 𝑧 ) + ( 𝑌 ‘ 𝑧 ) ) ≤ ( 𝐹 ‘ 𝑧 ) ↔ ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
51 |
49 50
|
bitr3d |
⊢ ( ( ( 𝑋 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝑌 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) ∈ ℝ ) → ( ( 𝑌 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ↔ ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
52 |
46 47 48 51
|
syl3an |
⊢ ( ( ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 ∧ ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ℕ0 ) → ( ( 𝑌 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ↔ ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
53 |
43 44 45 52
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( ( 𝑌 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ↔ ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
54 |
53
|
ralbidva |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( ∀ 𝑧 ∈ 𝐼 ( 𝑌 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝐼 ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
55 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ∈ V ) |
56 |
27
|
feqmptd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ) |
57 |
17
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 Fn 𝐼 ) |
58 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
59 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
60 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) = ( 𝑋 ‘ 𝑧 ) ) |
61 |
24 57 25 25 58 59 60
|
offval |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) ) |
62 |
25 44 55 56 61
|
ofrfval2 |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ↔ ∀ 𝑧 ∈ 𝐼 ( 𝑌 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) ) |
63 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ∈ V ) |
64 |
17
|
feqmptd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑋 ‘ 𝑧 ) ) ) |
65 |
27
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 Fn 𝐼 ) |
66 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑧 ) = ( 𝑌 ‘ 𝑧 ) ) |
67 |
24 65 25 25 58 59 66
|
offval |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑌 ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
68 |
25 43 63 64 67
|
ofrfval2 |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑋 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) ↔ ∀ 𝑧 ∈ 𝐼 ( 𝑋 ‘ 𝑧 ) ≤ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
69 |
54 62 68
|
3bitr4d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ↔ 𝑋 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) ) ) |
70 |
9 69
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) ) |
71 |
42 15 70
|
elrabd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) } ) |
72 |
41 71
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑌 ∈ 𝑆 ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) } ) ) |