Step |
Hyp |
Ref |
Expression |
1 |
|
prjsprel.1 |
⊢ ∼ = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ∃ 𝑙 ∈ 𝐾 𝑥 = ( 𝑙 · 𝑦 ) ) } |
2 |
|
prjspertr.b |
⊢ 𝐵 = ( ( Base ‘ 𝑉 ) ∖ { ( 0g ‘ 𝑉 ) } ) |
3 |
|
prjspertr.s |
⊢ 𝑆 = ( Scalar ‘ 𝑉 ) |
4 |
|
prjspertr.x |
⊢ · = ( ·𝑠 ‘ 𝑉 ) |
5 |
|
prjspertr.k |
⊢ 𝐾 = ( Base ‘ 𝑆 ) |
6 |
1
|
prjsprel |
⊢ ( 𝑋 ∼ 𝑌 ↔ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∃ 𝑚 ∈ 𝐾 𝑋 = ( 𝑚 · 𝑌 ) ) ) |
7 |
6
|
simprbi |
⊢ ( 𝑋 ∼ 𝑌 → ∃ 𝑚 ∈ 𝐾 𝑋 = ( 𝑚 · 𝑌 ) ) |
8 |
7
|
ad2antrl |
⊢ ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) → ∃ 𝑚 ∈ 𝐾 𝑋 = ( 𝑚 · 𝑌 ) ) |
9 |
|
simplrr |
⊢ ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) → 𝑌 ∼ 𝑍 ) |
10 |
1
|
prjsprel |
⊢ ( 𝑌 ∼ 𝑍 ↔ ( ( 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ ∃ 𝑛 ∈ 𝐾 𝑌 = ( 𝑛 · 𝑍 ) ) ) |
11 |
10
|
simprbi |
⊢ ( 𝑌 ∼ 𝑍 → ∃ 𝑛 ∈ 𝐾 𝑌 = ( 𝑛 · 𝑍 ) ) |
12 |
9 11
|
syl |
⊢ ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) → ∃ 𝑛 ∈ 𝐾 𝑌 = ( 𝑛 · 𝑍 ) ) |
13 |
|
simplrl |
⊢ ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) ) → 𝑋 ∼ 𝑌 ) |
14 |
13
|
anassrs |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑋 ∼ 𝑌 ) |
15 |
|
simpll |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∃ 𝑚 ∈ 𝐾 𝑋 = ( 𝑚 · 𝑌 ) ) → 𝑋 ∈ 𝐵 ) |
16 |
6 15
|
sylbi |
⊢ ( 𝑋 ∼ 𝑌 → 𝑋 ∈ 𝐵 ) |
17 |
14 16
|
syl |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑋 ∈ 𝐵 ) |
18 |
9
|
adantr |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑌 ∼ 𝑍 ) |
19 |
|
simplr |
⊢ ( ( ( 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ ∃ 𝑛 ∈ 𝐾 𝑌 = ( 𝑛 · 𝑍 ) ) → 𝑍 ∈ 𝐵 ) |
20 |
10 19
|
sylbi |
⊢ ( 𝑌 ∼ 𝑍 → 𝑍 ∈ 𝐵 ) |
21 |
18 20
|
syl |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑍 ∈ 𝐵 ) |
22 |
3
|
lmodring |
⊢ ( 𝑉 ∈ LMod → 𝑆 ∈ Ring ) |
23 |
22
|
ad3antrrr |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑆 ∈ Ring ) |
24 |
|
simplrl |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑚 ∈ 𝐾 ) |
25 |
|
simprl |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑛 ∈ 𝐾 ) |
26 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
27 |
5 26
|
ringcl |
⊢ ( ( 𝑆 ∈ Ring ∧ 𝑚 ∈ 𝐾 ∧ 𝑛 ∈ 𝐾 ) → ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) ∈ 𝐾 ) |
28 |
23 24 25 27
|
syl3anc |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) ∈ 𝐾 ) |
29 |
|
oveq1 |
⊢ ( 𝑜 = ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) → ( 𝑜 · 𝑍 ) = ( ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) · 𝑍 ) ) |
30 |
29
|
eqeq2d |
⊢ ( 𝑜 = ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) → ( 𝑋 = ( 𝑜 · 𝑍 ) ↔ 𝑋 = ( ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) · 𝑍 ) ) ) |
31 |
30
|
adantl |
⊢ ( ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) ∧ 𝑜 = ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) ) → ( 𝑋 = ( 𝑜 · 𝑍 ) ↔ 𝑋 = ( ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) · 𝑍 ) ) ) |
32 |
|
simprr |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑌 = ( 𝑛 · 𝑍 ) ) |
33 |
32
|
oveq2d |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → ( 𝑚 · 𝑌 ) = ( 𝑚 · ( 𝑛 · 𝑍 ) ) ) |
34 |
|
simplrr |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑋 = ( 𝑚 · 𝑌 ) ) |
35 |
|
simplll |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑉 ∈ LMod ) |
36 |
|
eldifi |
⊢ ( 𝑍 ∈ ( ( Base ‘ 𝑉 ) ∖ { ( 0g ‘ 𝑉 ) } ) → 𝑍 ∈ ( Base ‘ 𝑉 ) ) |
37 |
36 2
|
eleq2s |
⊢ ( 𝑍 ∈ 𝐵 → 𝑍 ∈ ( Base ‘ 𝑉 ) ) |
38 |
21 37
|
syl |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑍 ∈ ( Base ‘ 𝑉 ) ) |
39 |
|
eqid |
⊢ ( Base ‘ 𝑉 ) = ( Base ‘ 𝑉 ) |
40 |
39 3 4 5 26
|
lmodvsass |
⊢ ( ( 𝑉 ∈ LMod ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑛 ∈ 𝐾 ∧ 𝑍 ∈ ( Base ‘ 𝑉 ) ) ) → ( ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) · 𝑍 ) = ( 𝑚 · ( 𝑛 · 𝑍 ) ) ) |
41 |
35 24 25 38 40
|
syl13anc |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → ( ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) · 𝑍 ) = ( 𝑚 · ( 𝑛 · 𝑍 ) ) ) |
42 |
33 34 41
|
3eqtr4d |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑋 = ( ( 𝑚 ( .r ‘ 𝑆 ) 𝑛 ) · 𝑍 ) ) |
43 |
28 31 42
|
rspcedvd |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → ∃ 𝑜 ∈ 𝐾 𝑋 = ( 𝑜 · 𝑍 ) ) |
44 |
1
|
prjsprel |
⊢ ( 𝑋 ∼ 𝑍 ↔ ( ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ ∃ 𝑜 ∈ 𝐾 𝑋 = ( 𝑜 · 𝑍 ) ) ) |
45 |
17 21 43 44
|
syl21anbrc |
⊢ ( ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) ∧ ( 𝑛 ∈ 𝐾 ∧ 𝑌 = ( 𝑛 · 𝑍 ) ) ) → 𝑋 ∼ 𝑍 ) |
46 |
12 45
|
rexlimddv |
⊢ ( ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) ∧ ( 𝑚 ∈ 𝐾 ∧ 𝑋 = ( 𝑚 · 𝑌 ) ) ) → 𝑋 ∼ 𝑍 ) |
47 |
8 46
|
rexlimddv |
⊢ ( ( 𝑉 ∈ LMod ∧ ( 𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍 ) ) → 𝑋 ∼ 𝑍 ) |