Step |
Hyp |
Ref |
Expression |
1 |
|
mndind.ch |
⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) |
2 |
|
mndind.th |
⊢ ( 𝑥 = ( 𝑦 + 𝑧 ) → ( 𝜓 ↔ 𝜃 ) ) |
3 |
|
mndind.ta |
⊢ ( 𝑥 = 0 → ( 𝜓 ↔ 𝜏 ) ) |
4 |
|
mndind.et |
⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜂 ) ) |
5 |
|
mndind.0g |
⊢ 0 = ( 0g ‘ 𝑀 ) |
6 |
|
mndind.pg |
⊢ + = ( +g ‘ 𝑀 ) |
7 |
|
mndind.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
8 |
|
mndind.m |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
9 |
|
mndind.g |
⊢ ( 𝜑 → 𝐺 ⊆ 𝐵 ) |
10 |
|
mndind.k |
⊢ ( 𝜑 → 𝐵 = ( ( mrCls ‘ ( SubMnd ‘ 𝑀 ) ) ‘ 𝐺 ) ) |
11 |
|
mndind.i1 |
⊢ ( 𝜑 → 𝜏 ) |
12 |
|
mndind.i2 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺 ) ∧ 𝜒 ) → 𝜃 ) |
13 |
|
mndind.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
14 |
7 5
|
mndidcl |
⊢ ( 𝑀 ∈ Mnd → 0 ∈ 𝐵 ) |
15 |
8 14
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝐵 ) |
16 |
3
|
sbcieg |
⊢ ( 0 ∈ 𝐵 → ( [ 0 / 𝑥 ] 𝜓 ↔ 𝜏 ) ) |
17 |
15 16
|
syl |
⊢ ( 𝜑 → ( [ 0 / 𝑥 ] 𝜓 ↔ 𝜏 ) ) |
18 |
11 17
|
mpbird |
⊢ ( 𝜑 → [ 0 / 𝑥 ] 𝜓 ) |
19 |
|
dfsbcq |
⊢ ( 𝑎 = 0 → ( [ 𝑎 / 𝑥 ] 𝜓 ↔ [ 0 / 𝑥 ] 𝜓 ) ) |
20 |
|
oveq1 |
⊢ ( 𝑎 = 0 → ( 𝑎 + 𝐴 ) = ( 0 + 𝐴 ) ) |
21 |
20
|
sbceq1d |
⊢ ( 𝑎 = 0 → ( [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ↔ [ ( 0 + 𝐴 ) / 𝑥 ] 𝜓 ) ) |
22 |
19 21
|
imbi12d |
⊢ ( 𝑎 = 0 → ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ) ↔ ( [ 0 / 𝑥 ] 𝜓 → [ ( 0 + 𝐴 ) / 𝑥 ] 𝜓 ) ) ) |
23 |
7
|
submacs |
⊢ ( 𝑀 ∈ Mnd → ( SubMnd ‘ 𝑀 ) ∈ ( ACS ‘ 𝐵 ) ) |
24 |
8 23
|
syl |
⊢ ( 𝜑 → ( SubMnd ‘ 𝑀 ) ∈ ( ACS ‘ 𝐵 ) ) |
25 |
24
|
acsmred |
⊢ ( 𝜑 → ( SubMnd ‘ 𝑀 ) ∈ ( Moore ‘ 𝐵 ) ) |
26 |
|
eleq1w |
⊢ ( 𝑦 = 𝑎 → ( 𝑦 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵 ) ) |
27 |
26
|
anbi2d |
⊢ ( 𝑦 = 𝑎 → ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) ↔ ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑎 ∈ 𝐵 ) ) ) |
28 |
|
vex |
⊢ 𝑦 ∈ V |
29 |
28 1
|
sbcie |
⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) |
30 |
|
dfsbcq |
⊢ ( 𝑦 = 𝑎 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ [ 𝑎 / 𝑥 ] 𝜓 ) ) |
31 |
29 30
|
bitr3id |
⊢ ( 𝑦 = 𝑎 → ( 𝜒 ↔ [ 𝑎 / 𝑥 ] 𝜓 ) ) |
32 |
|
oveq1 |
⊢ ( 𝑦 = 𝑎 → ( 𝑦 + 𝑏 ) = ( 𝑎 + 𝑏 ) ) |
33 |
32
|
sbceq1d |
⊢ ( 𝑦 = 𝑎 → ( [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ) |
34 |
31 33
|
imbi12d |
⊢ ( 𝑦 = 𝑎 → ( ( 𝜒 → [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ) ) |
35 |
27 34
|
imbi12d |
⊢ ( 𝑦 = 𝑎 → ( ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝜒 → [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ) ) ↔ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑎 ∈ 𝐵 ) → ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ) ) ) |
36 |
|
eleq1w |
⊢ ( 𝑧 = 𝑏 → ( 𝑧 ∈ 𝐺 ↔ 𝑏 ∈ 𝐺 ) ) |
37 |
36
|
anbi2d |
⊢ ( 𝑧 = 𝑏 → ( ( 𝜑 ∧ 𝑧 ∈ 𝐺 ) ↔ ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ) ) |
38 |
37
|
anbi1d |
⊢ ( 𝑧 = 𝑏 → ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) ↔ ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) ) ) |
39 |
|
ovex |
⊢ ( 𝑦 + 𝑧 ) ∈ V |
40 |
39 2
|
sbcie |
⊢ ( [ ( 𝑦 + 𝑧 ) / 𝑥 ] 𝜓 ↔ 𝜃 ) |
41 |
|
oveq2 |
⊢ ( 𝑧 = 𝑏 → ( 𝑦 + 𝑧 ) = ( 𝑦 + 𝑏 ) ) |
42 |
41
|
sbceq1d |
⊢ ( 𝑧 = 𝑏 → ( [ ( 𝑦 + 𝑧 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ) ) |
43 |
40 42
|
bitr3id |
⊢ ( 𝑧 = 𝑏 → ( 𝜃 ↔ [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ) ) |
44 |
43
|
imbi2d |
⊢ ( 𝑧 = 𝑏 → ( ( 𝜒 → 𝜃 ) ↔ ( 𝜒 → [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ) ) ) |
45 |
38 44
|
imbi12d |
⊢ ( 𝑧 = 𝑏 → ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝜒 → 𝜃 ) ) ↔ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝜒 → [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ) ) ) ) |
46 |
12
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺 ) → ( 𝜒 → 𝜃 ) ) |
47 |
46
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 ∈ 𝐺 ) → ( 𝜒 → 𝜃 ) ) |
48 |
47
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝜒 → 𝜃 ) ) |
49 |
45 48
|
chvarvv |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝜒 → [ ( 𝑦 + 𝑏 ) / 𝑥 ] 𝜓 ) ) |
50 |
35 49
|
chvarvv |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) ∧ 𝑎 ∈ 𝐵 ) → ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ) |
51 |
50
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐺 ) → ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ) |
52 |
9 51
|
ssrabdv |
⊢ ( 𝜑 → 𝐺 ⊆ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ) |
53 |
7 6 5
|
mndrid |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑎 ∈ 𝐵 ) → ( 𝑎 + 0 ) = 𝑎 ) |
54 |
8 53
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐵 ) → ( 𝑎 + 0 ) = 𝑎 ) |
55 |
54
|
sbceq1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐵 ) → ( [ ( 𝑎 + 0 ) / 𝑥 ] 𝜓 ↔ [ 𝑎 / 𝑥 ] 𝜓 ) ) |
56 |
55
|
biimprd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐵 ) → ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 0 ) / 𝑥 ] 𝜓 ) ) |
57 |
56
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 0 ) / 𝑥 ] 𝜓 ) ) |
58 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ∧ ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ∧ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) ) ) → ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ) |
59 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) → 𝑀 ∈ Mnd ) |
60 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ 𝐵 ) |
61 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) → 𝑐 ∈ 𝐵 ) |
62 |
7 6
|
mndcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ) → ( 𝑏 + 𝑐 ) ∈ 𝐵 ) |
63 |
59 60 61 62
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝑏 + 𝑐 ) ∈ 𝐵 ) |
64 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑎 = ( 𝑏 + 𝑐 ) ) → 𝑎 = ( 𝑏 + 𝑐 ) ) |
65 |
64
|
sbceq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑎 = ( 𝑏 + 𝑐 ) ) → ( [ 𝑎 / 𝑥 ] 𝜓 ↔ [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 ) ) |
66 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝑏 + 𝑐 ) → ( 𝑎 + 𝑑 ) = ( ( 𝑏 + 𝑐 ) + 𝑑 ) ) |
67 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) → 𝑑 ∈ 𝐵 ) |
68 |
7 6
|
mndass |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) → ( ( 𝑏 + 𝑐 ) + 𝑑 ) = ( 𝑏 + ( 𝑐 + 𝑑 ) ) ) |
69 |
59 60 61 67 68
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) → ( ( 𝑏 + 𝑐 ) + 𝑑 ) = ( 𝑏 + ( 𝑐 + 𝑑 ) ) ) |
70 |
66 69
|
sylan9eqr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑎 = ( 𝑏 + 𝑐 ) ) → ( 𝑎 + 𝑑 ) = ( 𝑏 + ( 𝑐 + 𝑑 ) ) ) |
71 |
70
|
sbceq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑎 = ( 𝑏 + 𝑐 ) ) → ( [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
72 |
65 71
|
imbi12d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) ∧ 𝑎 = ( 𝑏 + 𝑐 ) ) → ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ↔ ( [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
73 |
63 72
|
rspcdv |
⊢ ( ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) ∧ 𝑏 ∈ 𝐵 ) → ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) → ( [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
74 |
73
|
ralrimdva |
⊢ ( ( 𝜑 ∧ ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ) → ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) → ∀ 𝑏 ∈ 𝐵 ( [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
75 |
74
|
impr |
⊢ ( ( 𝜑 ∧ ( ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ∧ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) ) → ∀ 𝑏 ∈ 𝐵 ( [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
76 |
|
oveq1 |
⊢ ( 𝑏 = 𝑎 → ( 𝑏 + 𝑐 ) = ( 𝑎 + 𝑐 ) ) |
77 |
76
|
sbceq1d |
⊢ ( 𝑏 = 𝑎 → ( [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ) |
78 |
|
oveq1 |
⊢ ( 𝑏 = 𝑎 → ( 𝑏 + ( 𝑐 + 𝑑 ) ) = ( 𝑎 + ( 𝑐 + 𝑑 ) ) ) |
79 |
78
|
sbceq1d |
⊢ ( 𝑏 = 𝑎 → ( [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
80 |
77 79
|
imbi12d |
⊢ ( 𝑏 = 𝑎 → ( ( [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ↔ ( [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
81 |
80
|
cbvralvw |
⊢ ( ∀ 𝑏 ∈ 𝐵 ( [ ( 𝑏 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑏 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ↔ ∀ 𝑎 ∈ 𝐵 ( [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
82 |
75 81
|
sylib |
⊢ ( ( 𝜑 ∧ ( ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ∧ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) ) → ∀ 𝑎 ∈ 𝐵 ( [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
83 |
82
|
adantrrl |
⊢ ( ( 𝜑 ∧ ( ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ∧ ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ∧ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) ) ) → ∀ 𝑎 ∈ 𝐵 ( [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
84 |
|
imim1 |
⊢ ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) → ( ( [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) → ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
85 |
84
|
ral2imi |
⊢ ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) → ( ∀ 𝑎 ∈ 𝐵 ( [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) → ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
86 |
58 83 85
|
sylc |
⊢ ( ( 𝜑 ∧ ( ( 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵 ) ∧ ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ∧ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) ) ) → ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
87 |
|
oveq2 |
⊢ ( 𝑏 = 0 → ( 𝑎 + 𝑏 ) = ( 𝑎 + 0 ) ) |
88 |
87
|
sbceq1d |
⊢ ( 𝑏 = 0 → ( [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + 0 ) / 𝑥 ] 𝜓 ) ) |
89 |
88
|
imbi2d |
⊢ ( 𝑏 = 0 → ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 0 ) / 𝑥 ] 𝜓 ) ) ) |
90 |
89
|
ralbidv |
⊢ ( 𝑏 = 0 → ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 0 ) / 𝑥 ] 𝜓 ) ) ) |
91 |
|
oveq2 |
⊢ ( 𝑏 = 𝑐 → ( 𝑎 + 𝑏 ) = ( 𝑎 + 𝑐 ) ) |
92 |
91
|
sbceq1d |
⊢ ( 𝑏 = 𝑐 → ( [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ) |
93 |
92
|
imbi2d |
⊢ ( 𝑏 = 𝑐 → ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ) ) |
94 |
93
|
ralbidv |
⊢ ( 𝑏 = 𝑐 → ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑐 ) / 𝑥 ] 𝜓 ) ) ) |
95 |
|
oveq2 |
⊢ ( 𝑏 = 𝑑 → ( 𝑎 + 𝑏 ) = ( 𝑎 + 𝑑 ) ) |
96 |
95
|
sbceq1d |
⊢ ( 𝑏 = 𝑑 → ( [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) |
97 |
96
|
imbi2d |
⊢ ( 𝑏 = 𝑑 → ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) ) |
98 |
97
|
ralbidv |
⊢ ( 𝑏 = 𝑑 → ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑑 ) / 𝑥 ] 𝜓 ) ) ) |
99 |
|
oveq2 |
⊢ ( 𝑏 = ( 𝑐 + 𝑑 ) → ( 𝑎 + 𝑏 ) = ( 𝑎 + ( 𝑐 + 𝑑 ) ) ) |
100 |
99
|
sbceq1d |
⊢ ( 𝑏 = ( 𝑐 + 𝑑 ) → ( [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) |
101 |
100
|
imbi2d |
⊢ ( 𝑏 = ( 𝑐 + 𝑑 ) → ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
102 |
101
|
ralbidv |
⊢ ( 𝑏 = ( 𝑐 + 𝑑 ) → ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + ( 𝑐 + 𝑑 ) ) / 𝑥 ] 𝜓 ) ) ) |
103 |
7 6 5 8 57 86 90 94 98 102
|
issubmd |
⊢ ( 𝜑 → { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ∈ ( SubMnd ‘ 𝑀 ) ) |
104 |
|
eqid |
⊢ ( mrCls ‘ ( SubMnd ‘ 𝑀 ) ) = ( mrCls ‘ ( SubMnd ‘ 𝑀 ) ) |
105 |
104
|
mrcsscl |
⊢ ( ( ( SubMnd ‘ 𝑀 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝐺 ⊆ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ∧ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ∈ ( SubMnd ‘ 𝑀 ) ) → ( ( mrCls ‘ ( SubMnd ‘ 𝑀 ) ) ‘ 𝐺 ) ⊆ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ) |
106 |
25 52 103 105
|
syl3anc |
⊢ ( 𝜑 → ( ( mrCls ‘ ( SubMnd ‘ 𝑀 ) ) ‘ 𝐺 ) ⊆ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ) |
107 |
10 106
|
eqsstrd |
⊢ ( 𝜑 → 𝐵 ⊆ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ) |
108 |
107 13
|
sseldd |
⊢ ( 𝜑 → 𝐴 ∈ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ) |
109 |
|
oveq2 |
⊢ ( 𝑏 = 𝐴 → ( 𝑎 + 𝑏 ) = ( 𝑎 + 𝐴 ) ) |
110 |
109
|
sbceq1d |
⊢ ( 𝑏 = 𝐴 → ( [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ↔ [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ) ) |
111 |
110
|
imbi2d |
⊢ ( 𝑏 = 𝐴 → ( ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ) ) ) |
112 |
111
|
ralbidv |
⊢ ( 𝑏 = 𝐴 → ( ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) ↔ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ) ) ) |
113 |
112
|
elrab |
⊢ ( 𝐴 ∈ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } ↔ ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ) ) ) |
114 |
113
|
simprbi |
⊢ ( 𝐴 ∈ { 𝑏 ∈ 𝐵 ∣ ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝑏 ) / 𝑥 ] 𝜓 ) } → ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ) ) |
115 |
108 114
|
syl |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐵 ( [ 𝑎 / 𝑥 ] 𝜓 → [ ( 𝑎 + 𝐴 ) / 𝑥 ] 𝜓 ) ) |
116 |
22 115 15
|
rspcdva |
⊢ ( 𝜑 → ( [ 0 / 𝑥 ] 𝜓 → [ ( 0 + 𝐴 ) / 𝑥 ] 𝜓 ) ) |
117 |
18 116
|
mpd |
⊢ ( 𝜑 → [ ( 0 + 𝐴 ) / 𝑥 ] 𝜓 ) |
118 |
7 6 5
|
mndlid |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( 0 + 𝐴 ) = 𝐴 ) |
119 |
8 13 118
|
syl2anc |
⊢ ( 𝜑 → ( 0 + 𝐴 ) = 𝐴 ) |
120 |
119
|
sbceq1d |
⊢ ( 𝜑 → ( [ ( 0 + 𝐴 ) / 𝑥 ] 𝜓 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) |
121 |
4
|
sbcieg |
⊢ ( 𝐴 ∈ 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜂 ) ) |
122 |
13 121
|
syl |
⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜂 ) ) |
123 |
120 122
|
bitrd |
⊢ ( 𝜑 → ( [ ( 0 + 𝐴 ) / 𝑥 ] 𝜓 ↔ 𝜂 ) ) |
124 |
117 123
|
mpbid |
⊢ ( 𝜑 → 𝜂 ) |