Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
psrbagconf1o.s |
⊢ 𝑆 = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } |
3 |
|
gsumbagdiagOLD.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
4 |
|
gsumbagdiagOLD.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
5 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) |
6 |
|
breq1 |
⊢ ( 𝑥 = 𝑌 → ( 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ↔ 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) ) |
7 |
6
|
elrab |
⊢ ( 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ↔ ( 𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) ) |
8 |
5 7
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑌 ∈ 𝐷 ∧ 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) ) |
9 |
8
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∈ 𝐷 ) |
10 |
8
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ) |
11 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐼 ∈ 𝑉 ) |
12 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐹 ∈ 𝐷 ) |
13 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∈ 𝑆 ) |
14 |
|
breq1 |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 ∘r ≤ 𝐹 ↔ 𝑋 ∘r ≤ 𝐹 ) ) |
15 |
14 2
|
elrab2 |
⊢ ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹 ) ) |
16 |
13 15
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹 ) ) |
17 |
16
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∈ 𝐷 ) |
18 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐷 ) → 𝑋 : 𝐼 ⟶ ℕ0 ) |
19 |
11 17 18
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 : 𝐼 ⟶ ℕ0 ) |
20 |
16
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∘r ≤ 𝐹 ) |
21 |
1
|
psrbagconOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝑋 : 𝐼 ⟶ ℕ0 ∧ 𝑋 ∘r ≤ 𝐹 ) ) → ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
22 |
11 12 19 20 21
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
23 |
22
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) |
24 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷 ) → 𝑌 : 𝐼 ⟶ ℕ0 ) |
25 |
11 9 24
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 : 𝐼 ⟶ ℕ0 ) |
26 |
22
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ) |
27 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ) → ( 𝐹 ∘f − 𝑋 ) : 𝐼 ⟶ ℕ0 ) |
28 |
11 26 27
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) : 𝐼 ⟶ ℕ0 ) |
29 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
30 |
11 12 29
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐹 : 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 |
11 25 28 30 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 |
10 23 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 |
9 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 |
19
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) ∈ ℕ0 ) |
44 |
25
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑧 ) ∈ ℕ0 ) |
45 |
30
|
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 |
25
|
feqmptd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑧 ) ) ) |
57 |
30
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝐹 Fn 𝐼 ) |
58 |
19
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 Fn 𝐼 ) |
59 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
60 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
61 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑧 ) = ( 𝑋 ‘ 𝑧 ) ) |
62 |
57 58 11 11 59 60 61
|
offval |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑋 ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑋 ‘ 𝑧 ) ) ) ) |
63 |
11 44 55 56 62
|
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 ) ) ) ) |
64 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ∈ V ) |
65 |
19
|
feqmptd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 = ( 𝑧 ∈ 𝐼 ↦ ( 𝑋 ‘ 𝑧 ) ) ) |
66 |
25
|
ffnd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑌 Fn 𝐼 ) |
67 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) ∧ 𝑧 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑧 ) = ( 𝑌 ‘ 𝑧 ) ) |
68 |
57 66 11 11 59 60 67
|
offval |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝐹 ∘f − 𝑌 ) = ( 𝑧 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑧 ) − ( 𝑌 ‘ 𝑧 ) ) ) ) |
69 |
11 43 64 65 68
|
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 ) ) ) ) |
70 |
54 63 69
|
3bitr4d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑌 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) ↔ 𝑋 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) ) ) |
71 |
10 70
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) ) |
72 |
42 17 71
|
elrabd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) } ) |
73 |
41 72
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑋 ) } ) ) → ( 𝑌 ∈ 𝑆 ∧ 𝑋 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑌 ) } ) ) |