Step |
Hyp |
Ref |
Expression |
1 |
|
iccshift.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
iccshift.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
iccshift.3 |
⊢ ( 𝜑 → 𝑇 ∈ ℝ ) |
4 |
|
eqeq1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 = ( 𝑧 + 𝑇 ) ↔ 𝑥 = ( 𝑧 + 𝑇 ) ) ) |
5 |
4
|
rexbidv |
⊢ ( 𝑤 = 𝑥 → ( ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑤 = ( 𝑧 + 𝑇 ) ↔ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) ) |
6 |
5
|
elrab |
⊢ ( 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑤 = ( 𝑧 + 𝑇 ) } ↔ ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) ) |
7 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) ) → ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) |
8 |
|
nfv |
⊢ Ⅎ 𝑧 𝜑 |
9 |
|
nfv |
⊢ Ⅎ 𝑧 𝑥 ∈ ℂ |
10 |
|
nfre1 |
⊢ Ⅎ 𝑧 ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) |
11 |
9 10
|
nfan |
⊢ Ⅎ 𝑧 ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) |
12 |
8 11
|
nfan |
⊢ Ⅎ 𝑧 ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) ) |
13 |
|
nfv |
⊢ Ⅎ 𝑧 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) |
14 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 = ( 𝑧 + 𝑇 ) ) |
15 |
1 2
|
iccssred |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
16 |
15
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ℝ ) |
17 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑇 ∈ ℝ ) |
18 |
16 17
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑧 + 𝑇 ) ∈ ℝ ) |
19 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ∈ ℝ ) |
20 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) |
21 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ ) |
22 |
|
elicc2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) ) |
23 |
19 21 22
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) ) |
24 |
20 23
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) |
25 |
24
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑧 ) |
26 |
19 16 17 25
|
leadd1dd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 + 𝑇 ) ≤ ( 𝑧 + 𝑇 ) ) |
27 |
24
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ≤ 𝐵 ) |
28 |
16 21 17 27
|
leadd1dd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑧 + 𝑇 ) ≤ ( 𝐵 + 𝑇 ) ) |
29 |
18 26 28
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝑧 + 𝑇 ) ∈ ℝ ∧ ( 𝐴 + 𝑇 ) ≤ ( 𝑧 + 𝑇 ) ∧ ( 𝑧 + 𝑇 ) ≤ ( 𝐵 + 𝑇 ) ) ) |
30 |
29
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( ( 𝑧 + 𝑇 ) ∈ ℝ ∧ ( 𝐴 + 𝑇 ) ≤ ( 𝑧 + 𝑇 ) ∧ ( 𝑧 + 𝑇 ) ≤ ( 𝐵 + 𝑇 ) ) ) |
31 |
1 3
|
readdcld |
⊢ ( 𝜑 → ( 𝐴 + 𝑇 ) ∈ ℝ ) |
32 |
31
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝐴 + 𝑇 ) ∈ ℝ ) |
33 |
2 3
|
readdcld |
⊢ ( 𝜑 → ( 𝐵 + 𝑇 ) ∈ ℝ ) |
34 |
33
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝐵 + 𝑇 ) ∈ ℝ ) |
35 |
|
elicc2 |
⊢ ( ( ( 𝐴 + 𝑇 ) ∈ ℝ ∧ ( 𝐵 + 𝑇 ) ∈ ℝ ) → ( ( 𝑧 + 𝑇 ) ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ↔ ( ( 𝑧 + 𝑇 ) ∈ ℝ ∧ ( 𝐴 + 𝑇 ) ≤ ( 𝑧 + 𝑇 ) ∧ ( 𝑧 + 𝑇 ) ≤ ( 𝐵 + 𝑇 ) ) ) ) |
36 |
32 34 35
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( ( 𝑧 + 𝑇 ) ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ↔ ( ( 𝑧 + 𝑇 ) ∈ ℝ ∧ ( 𝐴 + 𝑇 ) ≤ ( 𝑧 + 𝑇 ) ∧ ( 𝑧 + 𝑇 ) ≤ ( 𝐵 + 𝑇 ) ) ) ) |
37 |
30 36
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → ( 𝑧 + 𝑇 ) ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) |
38 |
14 37
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( 𝑧 + 𝑇 ) ) → 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) |
39 |
38
|
3exp |
⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝑥 = ( 𝑧 + 𝑇 ) → 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝑥 = ( 𝑧 + 𝑇 ) → 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) ) ) |
41 |
12 13 40
|
rexlimd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) ) → ( ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) → 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) ) |
42 |
7 41
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) ) → 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) |
43 |
6 42
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑤 = ( 𝑧 + 𝑇 ) } ) → 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) |
44 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝐴 + 𝑇 ) ∈ ℝ ) |
45 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝐵 + 𝑇 ) ∈ ℝ ) |
46 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) |
47 |
|
eliccre |
⊢ ( ( ( 𝐴 + 𝑇 ) ∈ ℝ ∧ ( 𝐵 + 𝑇 ) ∈ ℝ ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑥 ∈ ℝ ) |
48 |
44 45 46 47
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑥 ∈ ℝ ) |
49 |
48
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑥 ∈ ℂ ) |
50 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝐴 ∈ ℝ ) |
51 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝐵 ∈ ℝ ) |
52 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑇 ∈ ℝ ) |
53 |
48 52
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ℝ ) |
54 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
55 |
3
|
recnd |
⊢ ( 𝜑 → 𝑇 ∈ ℂ ) |
56 |
54 55
|
pncand |
⊢ ( 𝜑 → ( ( 𝐴 + 𝑇 ) − 𝑇 ) = 𝐴 ) |
57 |
56
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = ( ( 𝐴 + 𝑇 ) − 𝑇 ) ) |
58 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝐴 = ( ( 𝐴 + 𝑇 ) − 𝑇 ) ) |
59 |
|
elicc2 |
⊢ ( ( ( 𝐴 + 𝑇 ) ∈ ℝ ∧ ( 𝐵 + 𝑇 ) ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐴 + 𝑇 ) ≤ 𝑥 ∧ 𝑥 ≤ ( 𝐵 + 𝑇 ) ) ) ) |
60 |
44 45 59
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ↔ ( 𝑥 ∈ ℝ ∧ ( 𝐴 + 𝑇 ) ≤ 𝑥 ∧ 𝑥 ≤ ( 𝐵 + 𝑇 ) ) ) ) |
61 |
46 60
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝑥 ∈ ℝ ∧ ( 𝐴 + 𝑇 ) ≤ 𝑥 ∧ 𝑥 ≤ ( 𝐵 + 𝑇 ) ) ) |
62 |
61
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝐴 + 𝑇 ) ≤ 𝑥 ) |
63 |
44 48 52 62
|
lesub1dd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( ( 𝐴 + 𝑇 ) − 𝑇 ) ≤ ( 𝑥 − 𝑇 ) ) |
64 |
58 63
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝐴 ≤ ( 𝑥 − 𝑇 ) ) |
65 |
61
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑥 ≤ ( 𝐵 + 𝑇 ) ) |
66 |
48 45 52 65
|
lesub1dd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ≤ ( ( 𝐵 + 𝑇 ) − 𝑇 ) ) |
67 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
68 |
67 55
|
pncand |
⊢ ( 𝜑 → ( ( 𝐵 + 𝑇 ) − 𝑇 ) = 𝐵 ) |
69 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( ( 𝐵 + 𝑇 ) − 𝑇 ) = 𝐵 ) |
70 |
66 69
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ≤ 𝐵 ) |
71 |
50 51 53 64 70
|
eliccd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) ) |
72 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑇 ∈ ℂ ) |
73 |
49 72
|
npcand |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ( ( 𝑥 − 𝑇 ) + 𝑇 ) = 𝑥 ) |
74 |
73
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) |
75 |
|
oveq1 |
⊢ ( 𝑧 = ( 𝑥 − 𝑇 ) → ( 𝑧 + 𝑇 ) = ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) |
76 |
75
|
rspceeqv |
⊢ ( ( ( 𝑥 − 𝑇 ) ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 = ( ( 𝑥 − 𝑇 ) + 𝑇 ) ) → ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) |
77 |
71 74 76
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑥 = ( 𝑧 + 𝑇 ) ) |
78 |
49 77 6
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) → 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑤 = ( 𝑧 + 𝑇 ) } ) |
79 |
43 78
|
impbida |
⊢ ( 𝜑 → ( 𝑥 ∈ { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑤 = ( 𝑧 + 𝑇 ) } ↔ 𝑥 ∈ ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) ) |
80 |
79
|
eqrdv |
⊢ ( 𝜑 → { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑤 = ( 𝑧 + 𝑇 ) } = ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) ) |
81 |
80
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝐴 + 𝑇 ) [,] ( 𝐵 + 𝑇 ) ) = { 𝑤 ∈ ℂ ∣ ∃ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) 𝑤 = ( 𝑧 + 𝑇 ) } ) |