Step |
Hyp |
Ref |
Expression |
1 |
|
lsmssass.p |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
2 |
|
lsmssass.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
3 |
|
lsmssass.g |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
4 |
|
lsmssass.r |
⊢ ( 𝜑 → 𝑅 ⊆ 𝐵 ) |
5 |
|
lsmssass.t |
⊢ ( 𝜑 → 𝑇 ⊆ 𝐵 ) |
6 |
|
lsmssass.u |
⊢ ( 𝜑 → 𝑈 ⊆ 𝐵 ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
8 |
2 7 1
|
lsmvalx |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑅 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵 ) → ( 𝑅 ⊕ 𝑇 ) = ran ( 𝑎 ∈ 𝑅 , 𝑏 ∈ 𝑇 ↦ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) ) |
9 |
3 4 5 8
|
syl3anc |
⊢ ( 𝜑 → ( 𝑅 ⊕ 𝑇 ) = ran ( 𝑎 ∈ 𝑅 , 𝑏 ∈ 𝑇 ↦ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) ) |
10 |
9
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ( 𝑅 ⊕ 𝑇 ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ↔ ∃ 𝑦 ∈ ran ( 𝑎 ∈ 𝑅 , 𝑏 ∈ 𝑇 ↦ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
11 |
|
ovex |
⊢ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ V |
12 |
11
|
rgen2w |
⊢ ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑇 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ V |
13 |
|
eqid |
⊢ ( 𝑎 ∈ 𝑅 , 𝑏 ∈ 𝑇 ↦ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) = ( 𝑎 ∈ 𝑅 , 𝑏 ∈ 𝑇 ↦ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) |
14 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) |
15 |
14
|
eqeq2d |
⊢ ( 𝑦 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) → ( 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ↔ 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
16 |
15
|
rexbidv |
⊢ ( 𝑦 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) → ( ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ↔ ∃ 𝑐 ∈ 𝑈 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
17 |
13 16
|
rexrnmpo |
⊢ ( ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑇 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ V → ( ∃ 𝑦 ∈ ran ( 𝑎 ∈ 𝑅 , 𝑏 ∈ 𝑇 ↦ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ↔ ∃ 𝑎 ∈ 𝑅 ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
18 |
12 17
|
ax-mp |
⊢ ( ∃ 𝑦 ∈ ran ( 𝑎 ∈ 𝑅 , 𝑏 ∈ 𝑇 ↦ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ↔ ∃ 𝑎 ∈ 𝑅 ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) |
19 |
10 18
|
bitrdi |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ( 𝑅 ⊕ 𝑇 ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ↔ ∃ 𝑎 ∈ 𝑅 ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
20 |
2 7 1
|
lsmvalx |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑇 ⊕ 𝑈 ) = ran ( 𝑏 ∈ 𝑇 , 𝑐 ∈ 𝑈 ↦ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
21 |
3 5 6 20
|
syl3anc |
⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) = ran ( 𝑏 ∈ 𝑇 , 𝑐 ∈ 𝑈 ↦ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
22 |
21
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ ∃ 𝑧 ∈ ran ( 𝑏 ∈ 𝑇 , 𝑐 ∈ 𝑈 ↦ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
23 |
|
ovex |
⊢ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ∈ V |
24 |
23
|
rgen2w |
⊢ ∀ 𝑏 ∈ 𝑇 ∀ 𝑐 ∈ 𝑈 ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ∈ V |
25 |
|
eqid |
⊢ ( 𝑏 ∈ 𝑇 , 𝑐 ∈ 𝑈 ↦ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) = ( 𝑏 ∈ 𝑇 , 𝑐 ∈ 𝑈 ↦ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) |
26 |
|
oveq2 |
⊢ ( 𝑧 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
27 |
26
|
eqeq2d |
⊢ ( 𝑧 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) → ( 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) ) |
28 |
25 27
|
rexrnmpo |
⊢ ( ∀ 𝑏 ∈ 𝑇 ∀ 𝑐 ∈ 𝑈 ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ∈ V → ( ∃ 𝑧 ∈ ran ( 𝑏 ∈ 𝑇 , 𝑐 ∈ 𝑈 ↦ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) ) |
29 |
24 28
|
ax-mp |
⊢ ( ∃ 𝑧 ∈ ran ( 𝑏 ∈ 𝑇 , 𝑐 ∈ 𝑈 ↦ ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
30 |
22 29
|
bitrdi |
⊢ ( 𝜑 → ( ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) → ( ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) ) |
32 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝐺 ∈ Mnd ) |
33 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑅 ⊆ 𝐵 ) |
34 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑎 ∈ 𝑅 ) |
35 |
33 34
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑎 ∈ 𝐵 ) |
36 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑇 ⊆ 𝐵 ) |
37 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑏 ∈ 𝑇 ) |
38 |
36 37
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑏 ∈ 𝐵 ) |
39 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑈 ⊆ 𝐵 ) |
40 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑐 ∈ 𝑈 ) |
41 |
39 40
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → 𝑐 ∈ 𝐵 ) |
42 |
2 7
|
mndass |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) ) → ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
43 |
32 35 38 41 42
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
44 |
43
|
eqeq2d |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) ∧ ( 𝑏 ∈ 𝑇 ∧ 𝑐 ∈ 𝑈 ) ) → ( 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ↔ 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) ) |
45 |
44
|
2rexbidva |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) → ( ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ↔ ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) ( 𝑏 ( +g ‘ 𝐺 ) 𝑐 ) ) ) ) |
46 |
31 45
|
bitr4d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑅 ) → ( ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
47 |
46
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ 𝑅 ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ↔ ∃ 𝑎 ∈ 𝑅 ∃ 𝑏 ∈ 𝑇 ∃ 𝑐 ∈ 𝑈 𝑥 = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
48 |
19 47
|
bitr4d |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ( 𝑅 ⊕ 𝑇 ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ↔ ∃ 𝑎 ∈ 𝑅 ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
49 |
2 1
|
lsmssv |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑅 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵 ) → ( 𝑅 ⊕ 𝑇 ) ⊆ 𝐵 ) |
50 |
3 4 5 49
|
syl3anc |
⊢ ( 𝜑 → ( 𝑅 ⊕ 𝑇 ) ⊆ 𝐵 ) |
51 |
2 7 1
|
lsmelvalx |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑅 ⊕ 𝑇 ) ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑥 ∈ ( ( 𝑅 ⊕ 𝑇 ) ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ ( 𝑅 ⊕ 𝑇 ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
52 |
3 50 6 51
|
syl3anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝑅 ⊕ 𝑇 ) ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ ( 𝑅 ⊕ 𝑇 ) ∃ 𝑐 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑐 ) ) ) |
53 |
2 1
|
lsmssv |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑇 ⊕ 𝑈 ) ⊆ 𝐵 ) |
54 |
3 5 6 53
|
syl3anc |
⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) ⊆ 𝐵 ) |
55 |
2 7 1
|
lsmelvalx |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑅 ⊆ 𝐵 ∧ ( 𝑇 ⊕ 𝑈 ) ⊆ 𝐵 ) → ( 𝑥 ∈ ( 𝑅 ⊕ ( 𝑇 ⊕ 𝑈 ) ) ↔ ∃ 𝑎 ∈ 𝑅 ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
56 |
3 4 54 55
|
syl3anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑅 ⊕ ( 𝑇 ⊕ 𝑈 ) ) ↔ ∃ 𝑎 ∈ 𝑅 ∃ 𝑧 ∈ ( 𝑇 ⊕ 𝑈 ) 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
57 |
48 52 56
|
3bitr4d |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝑅 ⊕ 𝑇 ) ⊕ 𝑈 ) ↔ 𝑥 ∈ ( 𝑅 ⊕ ( 𝑇 ⊕ 𝑈 ) ) ) ) |
58 |
57
|
eqrdv |
⊢ ( 𝜑 → ( ( 𝑅 ⊕ 𝑇 ) ⊕ 𝑈 ) = ( 𝑅 ⊕ ( 𝑇 ⊕ 𝑈 ) ) ) |