Step |
Hyp |
Ref |
Expression |
1 |
|
rnghmsubcsetc.c |
⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 ) |
2 |
|
rnghmsubcsetc.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
3 |
|
rnghmsubcsetc.b |
⊢ ( 𝜑 → 𝐵 = ( Rng ∩ 𝑈 ) ) |
4 |
|
rnghmsubcsetc.h |
⊢ ( 𝜑 → 𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) ) |
5 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝜑 ) |
6 |
5
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝜑 ) |
7 |
6
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝜑 ) |
8 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) |
9 |
8
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) |
10 |
|
simpr |
⊢ ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) → 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) |
11 |
10
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) |
12 |
4
|
rnghmresel |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) → 𝑔 ∈ ( 𝑦 RngHomo 𝑧 ) ) |
13 |
7 9 11 12
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑔 ∈ ( 𝑦 RngHomo 𝑧 ) ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
15 |
|
simpl |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
16 |
14 15
|
anim12i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) |
17 |
16
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) |
18 |
|
simprl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) |
19 |
4
|
rnghmresel |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) → 𝑓 ∈ ( 𝑥 RngHomo 𝑦 ) ) |
20 |
7 17 18 19
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑓 ∈ ( 𝑥 RngHomo 𝑦 ) ) |
21 |
|
rnghmco |
⊢ ( ( 𝑔 ∈ ( 𝑦 RngHomo 𝑧 ) ∧ 𝑓 ∈ ( 𝑥 RngHomo 𝑦 ) ) → ( 𝑔 ∘ 𝑓 ) ∈ ( 𝑥 RngHomo 𝑧 ) ) |
22 |
13 20 21
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑔 ∘ 𝑓 ) ∈ ( 𝑥 RngHomo 𝑧 ) ) |
23 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑈 ∈ 𝑉 ) |
24 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
25 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ( Rng ∩ 𝑈 ) ) ) |
26 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( Rng ∩ 𝑈 ) → 𝑥 ∈ 𝑈 ) |
27 |
25 26
|
syl6bi |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈 ) ) |
28 |
27
|
imp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝑈 ) |
29 |
28
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑥 ∈ 𝑈 ) |
30 |
29
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑥 ∈ 𝑈 ) |
31 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ( Rng ∩ 𝑈 ) ) ) |
32 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( Rng ∩ 𝑈 ) → 𝑦 ∈ 𝑈 ) |
33 |
31 32
|
syl6bi |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈 ) ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈 ) ) |
35 |
34
|
com12 |
⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑦 ∈ 𝑈 ) ) |
36 |
35
|
adantr |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑦 ∈ 𝑈 ) ) |
37 |
36
|
impcom |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑦 ∈ 𝑈 ) |
38 |
37
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑦 ∈ 𝑈 ) |
39 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ ( Rng ∩ 𝑈 ) ) ) |
40 |
|
elinel2 |
⊢ ( 𝑧 ∈ ( Rng ∩ 𝑈 ) → 𝑧 ∈ 𝑈 ) |
41 |
39 40
|
syl6bi |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ 𝑈 ) ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ 𝑈 ) ) |
43 |
42
|
adantld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → 𝑧 ∈ 𝑈 ) ) |
44 |
43
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑧 ∈ 𝑈 ) |
45 |
44
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑧 ∈ 𝑈 ) |
46 |
|
eqid |
⊢ ( Base ‘ 𝑥 ) = ( Base ‘ 𝑥 ) |
47 |
|
eqid |
⊢ ( Base ‘ 𝑦 ) = ( Base ‘ 𝑦 ) |
48 |
|
eqid |
⊢ ( Base ‘ 𝑧 ) = ( Base ‘ 𝑧 ) |
49 |
|
simprl |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) → 𝜑 ) |
50 |
49
|
adantr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) ∧ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) → 𝜑 ) |
51 |
14
|
anim1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) |
52 |
51
|
ancoms |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) |
53 |
52
|
adantr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) ∧ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) |
54 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) ∧ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) → 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) |
55 |
50 53 54 19
|
syl3anc |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) ∧ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) → 𝑓 ∈ ( 𝑥 RngHomo 𝑦 ) ) |
56 |
46 47
|
rnghmf |
⊢ ( 𝑓 ∈ ( 𝑥 RngHomo 𝑦 ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) |
57 |
55 56
|
syl |
⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) ∧ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) |
58 |
57
|
ex |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) |
59 |
58
|
ex |
⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) ) |
60 |
59
|
adantr |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) ) |
61 |
60
|
impcom |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) |
62 |
61
|
com12 |
⊢ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) |
63 |
62
|
adantr |
⊢ ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) ) |
64 |
63
|
impcom |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑓 : ( Base ‘ 𝑥 ) ⟶ ( Base ‘ 𝑦 ) ) |
65 |
12
|
3expa |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) → 𝑔 ∈ ( 𝑦 RngHomo 𝑧 ) ) |
66 |
47 48
|
rnghmf |
⊢ ( 𝑔 ∈ ( 𝑦 RngHomo 𝑧 ) → 𝑔 : ( Base ‘ 𝑦 ) ⟶ ( Base ‘ 𝑧 ) ) |
67 |
65 66
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) → 𝑔 : ( Base ‘ 𝑦 ) ⟶ ( Base ‘ 𝑧 ) ) |
68 |
67
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) → 𝑔 : ( Base ‘ 𝑦 ) ⟶ ( Base ‘ 𝑧 ) ) ) |
69 |
68
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) → 𝑔 : ( Base ‘ 𝑦 ) ⟶ ( Base ‘ 𝑧 ) ) ) |
70 |
69
|
adantld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) → 𝑔 : ( Base ‘ 𝑦 ) ⟶ ( Base ‘ 𝑧 ) ) ) |
71 |
70
|
imp |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → 𝑔 : ( Base ‘ 𝑦 ) ⟶ ( Base ‘ 𝑧 ) ) |
72 |
1 23 24 30 38 45 46 47 48 64 71
|
estrcco |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ∘ 𝑓 ) ) |
73 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) ) |
74 |
73
|
oveqdr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 𝐻 𝑧 ) = ( 𝑥 ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) 𝑧 ) ) |
75 |
|
ovres |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑥 ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) 𝑧 ) = ( 𝑥 RngHomo 𝑧 ) ) |
76 |
75
|
ad2ant2l |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) 𝑧 ) = ( 𝑥 RngHomo 𝑧 ) ) |
77 |
74 76
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 𝐻 𝑧 ) = ( 𝑥 RngHomo 𝑧 ) ) |
78 |
77
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑥 𝐻 𝑧 ) = ( 𝑥 RngHomo 𝑧 ) ) |
79 |
22 72 78
|
3eltr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
80 |
79
|
ralrimivva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |
81 |
80
|
ralrimivva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) ) |