Metamath Proof Explorer


Theorem rnghmsubcsetclem2

Description: Lemma 2 for rnghmsubcsetc . (Contributed by AV, 9-Mar-2020)

Ref Expression
Hypotheses rnghmsubcsetc.c 𝐶 = ( ExtStrCat ‘ 𝑈 )
rnghmsubcsetc.u ( 𝜑𝑈𝑉 )
rnghmsubcsetc.b ( 𝜑𝐵 = ( Rng ∩ 𝑈 ) )
rnghmsubcsetc.h ( 𝜑𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )
Assertion rnghmsubcsetclem2 ( ( 𝜑𝑥𝐵 ) → ∀ 𝑦𝐵𝑧𝐵𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐻 𝑧 ) )

Proof

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