| Step |
Hyp |
Ref |
Expression |
| 1 |
|
addsproplem.1 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ) |
| 2 |
|
addspropord.2 |
⊢ ( 𝜑 → 𝑋 ∈ No ) |
| 3 |
|
addspropord.3 |
⊢ ( 𝜑 → 𝑌 ∈ No ) |
| 4 |
|
addspropord.4 |
⊢ ( 𝜑 → 𝑍 ∈ No ) |
| 5 |
|
addspropord.5 |
⊢ ( 𝜑 → 𝑌 <s 𝑍 ) |
| 6 |
|
addsproplem6.6 |
⊢ ( 𝜑 → ( bday ‘ 𝑌 ) = ( bday ‘ 𝑍 ) ) |
| 7 |
|
nodense |
⊢ ( ( ( 𝑌 ∈ No ∧ 𝑍 ∈ No ) ∧ ( ( bday ‘ 𝑌 ) = ( bday ‘ 𝑍 ) ∧ 𝑌 <s 𝑍 ) ) → ∃ 𝑚 ∈ No ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) |
| 8 |
3 4 6 5 7
|
syl22anc |
⊢ ( 𝜑 → ∃ 𝑚 ∈ No ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) |
| 9 |
1 2 3
|
addsproplem3 |
⊢ ( 𝜑 → ( ( 𝑋 +s 𝑌 ) ∈ No ∧ ( { 𝑎 ∣ ∃ 𝑏 ∈ ( L ‘ 𝑋 ) 𝑎 = ( 𝑏 +s 𝑌 ) } ∪ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑌 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑒 ∣ ∃ 𝑓 ∈ ( R ‘ 𝑋 ) 𝑒 = ( 𝑓 +s 𝑌 ) } ∪ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } ) ) ) |
| 10 |
9
|
simp1d |
⊢ ( 𝜑 → ( 𝑋 +s 𝑌 ) ∈ No ) |
| 11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑌 ) ∈ No ) |
| 12 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ) |
| 13 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑋 ∈ No ) |
| 14 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑚 ∈ No ) |
| 15 |
|
unidm |
⊢ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ) = ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) |
| 16 |
|
simprr1 |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ) |
| 17 |
|
bdayelon |
⊢ ( bday ‘ 𝑚 ) ∈ On |
| 18 |
|
bdayelon |
⊢ ( bday ‘ 𝑌 ) ∈ On |
| 19 |
|
bdayelon |
⊢ ( bday ‘ 𝑋 ) ∈ On |
| 20 |
|
naddel2 |
⊢ ( ( ( bday ‘ 𝑚 ) ∈ On ∧ ( bday ‘ 𝑌 ) ∈ On ∧ ( bday ‘ 𝑋 ) ∈ On ) → ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ↔ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∈ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) ) |
| 21 |
17 18 19 20
|
mp3an |
⊢ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ↔ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∈ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) |
| 22 |
16 21
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∈ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) |
| 23 |
|
elun1 |
⊢ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∈ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) → ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) ) |
| 24 |
22 23
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) ) |
| 25 |
15 24
|
eqeltrid |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑚 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) ) |
| 26 |
12 13 14 14 25
|
addsproplem1 |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( ( 𝑋 +s 𝑚 ) ∈ No ∧ ( 𝑚 <s 𝑚 → ( 𝑚 +s 𝑋 ) <s ( 𝑚 +s 𝑋 ) ) ) ) |
| 27 |
26
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑚 ) ∈ No ) |
| 28 |
|
uncom |
⊢ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) = ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) |
| 29 |
28
|
eleq2i |
⊢ ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) ↔ ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) ) |
| 30 |
29
|
imbi1i |
⊢ ( ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ) |
| 31 |
30
|
ralbii |
⊢ ( ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ↔ ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ) |
| 32 |
31
|
2ralbii |
⊢ ( ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ↔ ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ) |
| 33 |
1 32
|
sylib |
⊢ ( 𝜑 → ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ) |
| 34 |
33 2 4
|
addsproplem3 |
⊢ ( 𝜑 → ( ( 𝑋 +s 𝑍 ) ∈ No ∧ ( { 𝑎 ∣ ∃ 𝑏 ∈ ( L ‘ 𝑋 ) 𝑎 = ( 𝑏 +s 𝑍 ) } ∪ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ) <<s { ( 𝑋 +s 𝑍 ) } ∧ { ( 𝑋 +s 𝑍 ) } <<s ( { 𝑒 ∣ ∃ 𝑓 ∈ ( R ‘ 𝑋 ) 𝑒 = ( 𝑓 +s 𝑍 ) } ∪ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑍 ) 𝑔 = ( 𝑋 +s ℎ ) } ) ) ) |
| 35 |
34
|
simp1d |
⊢ ( 𝜑 → ( 𝑋 +s 𝑍 ) ∈ No ) |
| 36 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑍 ) ∈ No ) |
| 37 |
9
|
simp3d |
⊢ ( 𝜑 → { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑒 ∣ ∃ 𝑓 ∈ ( R ‘ 𝑋 ) 𝑒 = ( 𝑓 +s 𝑌 ) } ∪ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } ) ) |
| 38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑒 ∣ ∃ 𝑓 ∈ ( R ‘ 𝑋 ) 𝑒 = ( 𝑓 +s 𝑌 ) } ∪ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } ) ) |
| 39 |
|
ovex |
⊢ ( 𝑋 +s 𝑌 ) ∈ V |
| 40 |
39
|
snid |
⊢ ( 𝑋 +s 𝑌 ) ∈ { ( 𝑋 +s 𝑌 ) } |
| 41 |
40
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑌 ) ∈ { ( 𝑋 +s 𝑌 ) } ) |
| 42 |
|
oldbday |
⊢ ( ( ( bday ‘ 𝑌 ) ∈ On ∧ 𝑚 ∈ No ) → ( 𝑚 ∈ ( O ‘ ( bday ‘ 𝑌 ) ) ↔ ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ) ) |
| 43 |
18 14 42
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑚 ∈ ( O ‘ ( bday ‘ 𝑌 ) ) ↔ ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ) ) |
| 44 |
16 43
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑚 ∈ ( O ‘ ( bday ‘ 𝑌 ) ) ) |
| 45 |
|
simprr2 |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑌 <s 𝑚 ) |
| 46 |
|
elright |
⊢ ( 𝑚 ∈ ( R ‘ 𝑌 ) ↔ ( 𝑚 ∈ ( O ‘ ( bday ‘ 𝑌 ) ) ∧ 𝑌 <s 𝑚 ) ) |
| 47 |
44 45 46
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑚 ∈ ( R ‘ 𝑌 ) ) |
| 48 |
|
eqid |
⊢ ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑚 ) |
| 49 |
|
oveq2 |
⊢ ( ℎ = 𝑚 → ( 𝑋 +s ℎ ) = ( 𝑋 +s 𝑚 ) ) |
| 50 |
49
|
rspceeqv |
⊢ ( ( 𝑚 ∈ ( R ‘ 𝑌 ) ∧ ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑚 ) ) → ∃ ℎ ∈ ( R ‘ 𝑌 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s ℎ ) ) |
| 51 |
47 48 50
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ∃ ℎ ∈ ( R ‘ 𝑌 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s ℎ ) ) |
| 52 |
|
ovex |
⊢ ( 𝑋 +s 𝑚 ) ∈ V |
| 53 |
|
eqeq1 |
⊢ ( 𝑔 = ( 𝑋 +s 𝑚 ) → ( 𝑔 = ( 𝑋 +s ℎ ) ↔ ( 𝑋 +s 𝑚 ) = ( 𝑋 +s ℎ ) ) ) |
| 54 |
53
|
rexbidv |
⊢ ( 𝑔 = ( 𝑋 +s 𝑚 ) → ( ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) ↔ ∃ ℎ ∈ ( R ‘ 𝑌 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s ℎ ) ) ) |
| 55 |
52 54
|
elab |
⊢ ( ( 𝑋 +s 𝑚 ) ∈ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } ↔ ∃ ℎ ∈ ( R ‘ 𝑌 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s ℎ ) ) |
| 56 |
51 55
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑚 ) ∈ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } ) |
| 57 |
|
elun2 |
⊢ ( ( 𝑋 +s 𝑚 ) ∈ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } → ( 𝑋 +s 𝑚 ) ∈ ( { 𝑒 ∣ ∃ 𝑓 ∈ ( R ‘ 𝑋 ) 𝑒 = ( 𝑓 +s 𝑌 ) } ∪ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } ) ) |
| 58 |
56 57
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑚 ) ∈ ( { 𝑒 ∣ ∃ 𝑓 ∈ ( R ‘ 𝑋 ) 𝑒 = ( 𝑓 +s 𝑌 ) } ∪ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑌 ) 𝑔 = ( 𝑋 +s ℎ ) } ) ) |
| 59 |
38 41 58
|
ssltsepcd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑌 ) <s ( 𝑋 +s 𝑚 ) ) |
| 60 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ∀ 𝑥 ∈ No ∀ 𝑦 ∈ No ∀ 𝑧 ∈ No ( ( ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑦 ) ) ∪ ( ( bday ‘ 𝑥 ) +no ( bday ‘ 𝑧 ) ) ) ∈ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑍 ) ) ∪ ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) → ( ( 𝑥 +s 𝑦 ) ∈ No ∧ ( 𝑦 <s 𝑧 → ( 𝑦 +s 𝑥 ) <s ( 𝑧 +s 𝑥 ) ) ) ) ) |
| 61 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑍 ∈ No ) |
| 62 |
60 13 61
|
addsproplem3 |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( ( 𝑋 +s 𝑍 ) ∈ No ∧ ( { 𝑎 ∣ ∃ 𝑏 ∈ ( L ‘ 𝑋 ) 𝑎 = ( 𝑏 +s 𝑍 ) } ∪ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ) <<s { ( 𝑋 +s 𝑍 ) } ∧ { ( 𝑋 +s 𝑍 ) } <<s ( { 𝑒 ∣ ∃ 𝑓 ∈ ( R ‘ 𝑋 ) 𝑒 = ( 𝑓 +s 𝑍 ) } ∪ { 𝑔 ∣ ∃ ℎ ∈ ( R ‘ 𝑍 ) 𝑔 = ( 𝑋 +s ℎ ) } ) ) ) |
| 63 |
62
|
simp2d |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( { 𝑎 ∣ ∃ 𝑏 ∈ ( L ‘ 𝑋 ) 𝑎 = ( 𝑏 +s 𝑍 ) } ∪ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ) <<s { ( 𝑋 +s 𝑍 ) } ) |
| 64 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( bday ‘ 𝑌 ) = ( bday ‘ 𝑍 ) ) |
| 65 |
16 64
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑍 ) ) |
| 66 |
|
bdayelon |
⊢ ( bday ‘ 𝑍 ) ∈ On |
| 67 |
|
oldbday |
⊢ ( ( ( bday ‘ 𝑍 ) ∈ On ∧ 𝑚 ∈ No ) → ( 𝑚 ∈ ( O ‘ ( bday ‘ 𝑍 ) ) ↔ ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑍 ) ) ) |
| 68 |
66 14 67
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑚 ∈ ( O ‘ ( bday ‘ 𝑍 ) ) ↔ ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑍 ) ) ) |
| 69 |
65 68
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑚 ∈ ( O ‘ ( bday ‘ 𝑍 ) ) ) |
| 70 |
|
simprr3 |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑚 <s 𝑍 ) |
| 71 |
|
elleft |
⊢ ( 𝑚 ∈ ( L ‘ 𝑍 ) ↔ ( 𝑚 ∈ ( O ‘ ( bday ‘ 𝑍 ) ) ∧ 𝑚 <s 𝑍 ) ) |
| 72 |
69 70 71
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → 𝑚 ∈ ( L ‘ 𝑍 ) ) |
| 73 |
|
oveq2 |
⊢ ( 𝑑 = 𝑚 → ( 𝑋 +s 𝑑 ) = ( 𝑋 +s 𝑚 ) ) |
| 74 |
73
|
rspceeqv |
⊢ ( ( 𝑚 ∈ ( L ‘ 𝑍 ) ∧ ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑚 ) ) → ∃ 𝑑 ∈ ( L ‘ 𝑍 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑑 ) ) |
| 75 |
72 48 74
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ∃ 𝑑 ∈ ( L ‘ 𝑍 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑑 ) ) |
| 76 |
|
eqeq1 |
⊢ ( 𝑐 = ( 𝑋 +s 𝑚 ) → ( 𝑐 = ( 𝑋 +s 𝑑 ) ↔ ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑑 ) ) ) |
| 77 |
76
|
rexbidv |
⊢ ( 𝑐 = ( 𝑋 +s 𝑚 ) → ( ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) ↔ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑑 ) ) ) |
| 78 |
52 77
|
elab |
⊢ ( ( 𝑋 +s 𝑚 ) ∈ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ↔ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) ( 𝑋 +s 𝑚 ) = ( 𝑋 +s 𝑑 ) ) |
| 79 |
75 78
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑚 ) ∈ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ) |
| 80 |
|
elun2 |
⊢ ( ( 𝑋 +s 𝑚 ) ∈ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } → ( 𝑋 +s 𝑚 ) ∈ ( { 𝑎 ∣ ∃ 𝑏 ∈ ( L ‘ 𝑋 ) 𝑎 = ( 𝑏 +s 𝑍 ) } ∪ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ) ) |
| 81 |
79 80
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑚 ) ∈ ( { 𝑎 ∣ ∃ 𝑏 ∈ ( L ‘ 𝑋 ) 𝑎 = ( 𝑏 +s 𝑍 ) } ∪ { 𝑐 ∣ ∃ 𝑑 ∈ ( L ‘ 𝑍 ) 𝑐 = ( 𝑋 +s 𝑑 ) } ) ) |
| 82 |
|
ovex |
⊢ ( 𝑋 +s 𝑍 ) ∈ V |
| 83 |
82
|
snid |
⊢ ( 𝑋 +s 𝑍 ) ∈ { ( 𝑋 +s 𝑍 ) } |
| 84 |
83
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑍 ) ∈ { ( 𝑋 +s 𝑍 ) } ) |
| 85 |
63 81 84
|
ssltsepcd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑚 ) <s ( 𝑋 +s 𝑍 ) ) |
| 86 |
11 27 36 59 85
|
slttrd |
⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ No ∧ ( ( bday ‘ 𝑚 ) ∈ ( bday ‘ 𝑌 ) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍 ) ) ) → ( 𝑋 +s 𝑌 ) <s ( 𝑋 +s 𝑍 ) ) |
| 87 |
8 86
|
rexlimddv |
⊢ ( 𝜑 → ( 𝑋 +s 𝑌 ) <s ( 𝑋 +s 𝑍 ) ) |
| 88 |
3 2
|
addscomd |
⊢ ( 𝜑 → ( 𝑌 +s 𝑋 ) = ( 𝑋 +s 𝑌 ) ) |
| 89 |
4 2
|
addscomd |
⊢ ( 𝜑 → ( 𝑍 +s 𝑋 ) = ( 𝑋 +s 𝑍 ) ) |
| 90 |
87 88 89
|
3brtr4d |
⊢ ( 𝜑 → ( 𝑌 +s 𝑋 ) <s ( 𝑍 +s 𝑋 ) ) |