Step |
Hyp |
Ref |
Expression |
1 |
|
ltexprlem.1 |
⊢ 𝐶 = { 𝑥 ∣ ∃ 𝑦 ( ¬ 𝑦 ∈ 𝐴 ∧ ( 𝑦 +Q 𝑥 ) ∈ 𝐵 ) } |
2 |
1
|
ltexprlem5 |
⊢ ( ( 𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵 ) → 𝐶 ∈ P ) |
3 |
|
ltaddpr |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝐴 <P ( 𝐴 +P 𝐶 ) ) |
4 |
|
addclpr |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝐴 +P 𝐶 ) ∈ P ) |
5 |
|
ltprord |
⊢ ( ( 𝐴 ∈ P ∧ ( 𝐴 +P 𝐶 ) ∈ P ) → ( 𝐴 <P ( 𝐴 +P 𝐶 ) ↔ 𝐴 ⊊ ( 𝐴 +P 𝐶 ) ) ) |
6 |
4 5
|
syldan |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝐴 <P ( 𝐴 +P 𝐶 ) ↔ 𝐴 ⊊ ( 𝐴 +P 𝐶 ) ) ) |
7 |
3 6
|
mpbid |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝐴 ⊊ ( 𝐴 +P 𝐶 ) ) |
8 |
7
|
pssssd |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝐴 ⊆ ( 𝐴 +P 𝐶 ) ) |
9 |
8
|
sseld |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝑤 ∈ 𝐴 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) |
10 |
9
|
2a1d |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → ( 𝑤 ∈ 𝐴 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
11 |
10
|
com4r |
⊢ ( 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
12 |
11
|
expd |
⊢ ( 𝑤 ∈ 𝐴 → ( 𝐴 ∈ P → ( 𝐶 ∈ P → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) ) |
13 |
|
prnmadd |
⊢ ( ( 𝐵 ∈ P ∧ 𝑤 ∈ 𝐵 ) → ∃ 𝑣 ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) |
14 |
13
|
ex |
⊢ ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → ∃ 𝑣 ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) ) |
15 |
|
elprnq |
⊢ ( ( 𝐵 ∈ P ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) → ( 𝑤 +Q 𝑣 ) ∈ Q ) |
16 |
|
addnqf |
⊢ +Q : ( Q × Q ) ⟶ Q |
17 |
16
|
fdmi |
⊢ dom +Q = ( Q × Q ) |
18 |
|
0nnq |
⊢ ¬ ∅ ∈ Q |
19 |
17 18
|
ndmovrcl |
⊢ ( ( 𝑤 +Q 𝑣 ) ∈ Q → ( 𝑤 ∈ Q ∧ 𝑣 ∈ Q ) ) |
20 |
15 19
|
syl |
⊢ ( ( 𝐵 ∈ P ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) → ( 𝑤 ∈ Q ∧ 𝑣 ∈ Q ) ) |
21 |
20
|
simpld |
⊢ ( ( 𝐵 ∈ P ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) → 𝑤 ∈ Q ) |
22 |
|
vex |
⊢ 𝑣 ∈ V |
23 |
22
|
prlem934 |
⊢ ( 𝐴 ∈ P → ∃ 𝑧 ∈ 𝐴 ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) |
24 |
23
|
adantr |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ∃ 𝑧 ∈ 𝐴 ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) |
25 |
|
prub |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑤 ∈ Q ) → ( ¬ 𝑤 ∈ 𝐴 → 𝑧 <Q 𝑤 ) ) |
26 |
|
ltexnq |
⊢ ( 𝑤 ∈ Q → ( 𝑧 <Q 𝑤 ↔ ∃ 𝑥 ( 𝑧 +Q 𝑥 ) = 𝑤 ) ) |
27 |
26
|
adantl |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑤 ∈ Q ) → ( 𝑧 <Q 𝑤 ↔ ∃ 𝑥 ( 𝑧 +Q 𝑥 ) = 𝑤 ) ) |
28 |
25 27
|
sylibd |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑤 ∈ Q ) → ( ¬ 𝑤 ∈ 𝐴 → ∃ 𝑥 ( 𝑧 +Q 𝑥 ) = 𝑤 ) ) |
29 |
28
|
ex |
⊢ ( ( 𝐴 ∈ P ∧ 𝑧 ∈ 𝐴 ) → ( 𝑤 ∈ Q → ( ¬ 𝑤 ∈ 𝐴 → ∃ 𝑥 ( 𝑧 +Q 𝑥 ) = 𝑤 ) ) ) |
30 |
29
|
ad2ant2r |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) → ( 𝑤 ∈ Q → ( ¬ 𝑤 ∈ 𝐴 → ∃ 𝑥 ( 𝑧 +Q 𝑥 ) = 𝑤 ) ) ) |
31 |
|
vex |
⊢ 𝑧 ∈ V |
32 |
|
vex |
⊢ 𝑥 ∈ V |
33 |
|
addcomnq |
⊢ ( 𝑓 +Q 𝑔 ) = ( 𝑔 +Q 𝑓 ) |
34 |
|
addassnq |
⊢ ( ( 𝑓 +Q 𝑔 ) +Q ℎ ) = ( 𝑓 +Q ( 𝑔 +Q ℎ ) ) |
35 |
31 22 32 33 34
|
caov32 |
⊢ ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) = ( ( 𝑧 +Q 𝑥 ) +Q 𝑣 ) |
36 |
|
oveq1 |
⊢ ( ( 𝑧 +Q 𝑥 ) = 𝑤 → ( ( 𝑧 +Q 𝑥 ) +Q 𝑣 ) = ( 𝑤 +Q 𝑣 ) ) |
37 |
35 36
|
eqtrid |
⊢ ( ( 𝑧 +Q 𝑥 ) = 𝑤 → ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) = ( 𝑤 +Q 𝑣 ) ) |
38 |
37
|
eleq1d |
⊢ ( ( 𝑧 +Q 𝑥 ) = 𝑤 → ( ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) ∈ 𝐵 ↔ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) ) |
39 |
38
|
biimpar |
⊢ ( ( ( 𝑧 +Q 𝑥 ) = 𝑤 ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) → ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) ∈ 𝐵 ) |
40 |
|
ovex |
⊢ ( 𝑧 +Q 𝑣 ) ∈ V |
41 |
|
eleq1 |
⊢ ( 𝑦 = ( 𝑧 +Q 𝑣 ) → ( 𝑦 ∈ 𝐴 ↔ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) |
42 |
41
|
notbid |
⊢ ( 𝑦 = ( 𝑧 +Q 𝑣 ) → ( ¬ 𝑦 ∈ 𝐴 ↔ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) |
43 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑧 +Q 𝑣 ) → ( 𝑦 +Q 𝑥 ) = ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) ) |
44 |
43
|
eleq1d |
⊢ ( 𝑦 = ( 𝑧 +Q 𝑣 ) → ( ( 𝑦 +Q 𝑥 ) ∈ 𝐵 ↔ ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) ∈ 𝐵 ) ) |
45 |
42 44
|
anbi12d |
⊢ ( 𝑦 = ( 𝑧 +Q 𝑣 ) → ( ( ¬ 𝑦 ∈ 𝐴 ∧ ( 𝑦 +Q 𝑥 ) ∈ 𝐵 ) ↔ ( ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ∧ ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) ∈ 𝐵 ) ) ) |
46 |
40 45
|
spcev |
⊢ ( ( ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ∧ ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) ∈ 𝐵 ) → ∃ 𝑦 ( ¬ 𝑦 ∈ 𝐴 ∧ ( 𝑦 +Q 𝑥 ) ∈ 𝐵 ) ) |
47 |
1
|
abeq2i |
⊢ ( 𝑥 ∈ 𝐶 ↔ ∃ 𝑦 ( ¬ 𝑦 ∈ 𝐴 ∧ ( 𝑦 +Q 𝑥 ) ∈ 𝐵 ) ) |
48 |
46 47
|
sylibr |
⊢ ( ( ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ∧ ( ( 𝑧 +Q 𝑣 ) +Q 𝑥 ) ∈ 𝐵 ) → 𝑥 ∈ 𝐶 ) |
49 |
39 48
|
sylan2 |
⊢ ( ( ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ∧ ( ( 𝑧 +Q 𝑥 ) = 𝑤 ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) ) → 𝑥 ∈ 𝐶 ) |
50 |
|
df-plp |
⊢ +P = ( 𝑥 ∈ P , 𝑤 ∈ P ↦ { 𝑧 ∣ ∃ 𝑓 ∈ 𝑥 ∃ 𝑣 ∈ 𝑤 𝑧 = ( 𝑓 +Q 𝑣 ) } ) |
51 |
|
addclnq |
⊢ ( ( 𝑓 ∈ Q ∧ 𝑣 ∈ Q ) → ( 𝑓 +Q 𝑣 ) ∈ Q ) |
52 |
50 51
|
genpprecl |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( ( 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶 ) → ( 𝑧 +Q 𝑥 ) ∈ ( 𝐴 +P 𝐶 ) ) ) |
53 |
49 52
|
sylan2i |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( ( 𝑧 ∈ 𝐴 ∧ ( ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ∧ ( ( 𝑧 +Q 𝑥 ) = 𝑤 ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) ) ) → ( 𝑧 +Q 𝑥 ) ∈ ( 𝐴 +P 𝐶 ) ) ) |
54 |
53
|
exp4d |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝑧 ∈ 𝐴 → ( ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 → ( ( ( 𝑧 +Q 𝑥 ) = 𝑤 ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) → ( 𝑧 +Q 𝑥 ) ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
55 |
54
|
imp42 |
⊢ ( ( ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) ∧ ( ( 𝑧 +Q 𝑥 ) = 𝑤 ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) ) → ( 𝑧 +Q 𝑥 ) ∈ ( 𝐴 +P 𝐶 ) ) |
56 |
|
eleq1 |
⊢ ( ( 𝑧 +Q 𝑥 ) = 𝑤 → ( ( 𝑧 +Q 𝑥 ) ∈ ( 𝐴 +P 𝐶 ) ↔ 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) |
57 |
56
|
ad2antrl |
⊢ ( ( ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) ∧ ( ( 𝑧 +Q 𝑥 ) = 𝑤 ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) ) → ( ( 𝑧 +Q 𝑥 ) ∈ ( 𝐴 +P 𝐶 ) ↔ 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) |
58 |
55 57
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) ∧ ( ( 𝑧 +Q 𝑥 ) = 𝑤 ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) ) → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) |
59 |
58
|
exp32 |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) → ( ( 𝑧 +Q 𝑥 ) = 𝑤 → ( ( 𝑤 +Q 𝑣 ) ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) |
60 |
59
|
exlimdv |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) → ( ∃ 𝑥 ( 𝑧 +Q 𝑥 ) = 𝑤 → ( ( 𝑤 +Q 𝑣 ) ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) |
61 |
30 60
|
syl6d |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ ( 𝑧 +Q 𝑣 ) ∈ 𝐴 ) ) → ( 𝑤 ∈ Q → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝑤 +Q 𝑣 ) ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
62 |
24 61
|
rexlimddv |
⊢ ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝑤 ∈ Q → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝑤 +Q 𝑣 ) ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
63 |
62
|
com14 |
⊢ ( ( 𝑤 +Q 𝑣 ) ∈ 𝐵 → ( 𝑤 ∈ Q → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
64 |
63
|
adantl |
⊢ ( ( 𝐵 ∈ P ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) → ( 𝑤 ∈ Q → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
65 |
21 64
|
mpd |
⊢ ( ( 𝐵 ∈ P ∧ ( 𝑤 +Q 𝑣 ) ∈ 𝐵 ) → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) |
66 |
65
|
ex |
⊢ ( 𝐵 ∈ P → ( ( 𝑤 +Q 𝑣 ) ∈ 𝐵 → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
67 |
66
|
exlimdv |
⊢ ( 𝐵 ∈ P → ( ∃ 𝑣 ( 𝑤 +Q 𝑣 ) ∈ 𝐵 → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
68 |
14 67
|
syld |
⊢ ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
69 |
68
|
com4t |
⊢ ( ¬ 𝑤 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐶 ∈ P ) → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
70 |
69
|
expd |
⊢ ( ¬ 𝑤 ∈ 𝐴 → ( 𝐴 ∈ P → ( 𝐶 ∈ P → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) ) |
71 |
12 70
|
pm2.61i |
⊢ ( 𝐴 ∈ P → ( 𝐶 ∈ P → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
72 |
2 71
|
syl5 |
⊢ ( 𝐴 ∈ P → ( ( 𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵 ) → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
73 |
72
|
expd |
⊢ ( 𝐴 ∈ P → ( 𝐵 ∈ P → ( 𝐴 ⊊ 𝐵 → ( 𝐵 ∈ P → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) ) |
74 |
73
|
com34 |
⊢ ( 𝐴 ∈ P → ( 𝐵 ∈ P → ( 𝐵 ∈ P → ( 𝐴 ⊊ 𝐵 → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) ) |
75 |
74
|
pm2.43d |
⊢ ( 𝐴 ∈ P → ( 𝐵 ∈ P → ( 𝐴 ⊊ 𝐵 → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) ) ) |
76 |
75
|
imp31 |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) ∧ 𝐴 ⊊ 𝐵 ) → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ ( 𝐴 +P 𝐶 ) ) ) |
77 |
76
|
ssrdv |
⊢ ( ( ( 𝐴 ∈ P ∧ 𝐵 ∈ P ) ∧ 𝐴 ⊊ 𝐵 ) → 𝐵 ⊆ ( 𝐴 +P 𝐶 ) ) |