Step | Hyp | Ref
| Expression |
1 | | mclsppslem.10 |
. . . 4
⊢ (𝜑 → 𝑠 ∈ ran 𝐿) |
2 | | mclspps.l |
. . . . 5
⊢ 𝐿 = (mSubst‘𝑇) |
3 | | mclspps.e |
. . . . 5
⊢ 𝐸 = (mEx‘𝑇) |
4 | 2, 3 | msubf 33237 |
. . . 4
⊢ (𝑠 ∈ ran 𝐿 → 𝑠:𝐸⟶𝐸) |
5 | 1, 4 | syl 17 |
. . 3
⊢ (𝜑 → 𝑠:𝐸⟶𝐸) |
6 | | mclspps.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ mFS) |
7 | | eqid 2739 |
. . . . . . . . 9
⊢
(mAx‘𝑇) =
(mAx‘𝑇) |
8 | | eqid 2739 |
. . . . . . . . 9
⊢
(mStat‘𝑇) =
(mStat‘𝑇) |
9 | 7, 8 | maxsta 33259 |
. . . . . . . 8
⊢ (𝑇 ∈ mFS →
(mAx‘𝑇) ⊆
(mStat‘𝑇)) |
10 | 6, 9 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (mAx‘𝑇) ⊆ (mStat‘𝑇)) |
11 | | eqid 2739 |
. . . . . . . 8
⊢
(mPreSt‘𝑇) =
(mPreSt‘𝑇) |
12 | 11, 8 | mstapst 33252 |
. . . . . . 7
⊢
(mStat‘𝑇)
⊆ (mPreSt‘𝑇) |
13 | 10, 12 | sstrdi 3929 |
. . . . . 6
⊢ (𝜑 → (mAx‘𝑇) ⊆ (mPreSt‘𝑇)) |
14 | | mclsppslem.9 |
. . . . . 6
⊢ (𝜑 → 〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇)) |
15 | 13, 14 | sseldd 3918 |
. . . . 5
⊢ (𝜑 → 〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇)) |
16 | | mclspps.d |
. . . . . 6
⊢ 𝐷 = (mDV‘𝑇) |
17 | 16, 3, 11 | elmpst 33241 |
. . . . 5
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸)) |
18 | 15, 17 | sylib 221 |
. . . 4
⊢ (𝜑 → ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸)) |
19 | 18 | simp3d 1146 |
. . 3
⊢ (𝜑 → 𝑝 ∈ 𝐸) |
20 | 5, 19 | ffvelrnd 6926 |
. 2
⊢ (𝜑 → (𝑠‘𝑝) ∈ 𝐸) |
21 | | fvco3 6831 |
. . . 4
⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝))) |
22 | 5, 19, 21 | syl2anc 587 |
. . 3
⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝))) |
23 | | mclspps.c |
. . . 4
⊢ 𝐶 = (mCls‘𝑇) |
24 | | mclspps.2 |
. . . 4
⊢ (𝜑 → 𝐾 ⊆ 𝐷) |
25 | | mclspps.3 |
. . . 4
⊢ (𝜑 → 𝐵 ⊆ 𝐸) |
26 | | mclspps.v |
. . . 4
⊢ 𝑉 = (mVR‘𝑇) |
27 | | mclspps.h |
. . . 4
⊢ 𝐻 = (mVH‘𝑇) |
28 | | mclspps.w |
. . . 4
⊢ 𝑊 = (mVars‘𝑇) |
29 | | mclspps.5 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ ran 𝐿) |
30 | 2 | msubco 33236 |
. . . . 5
⊢ ((𝑆 ∈ ran 𝐿 ∧ 𝑠 ∈ ran 𝐿) → (𝑆 ∘ 𝑠) ∈ ran 𝐿) |
31 | 29, 1, 30 | syl2anc 587 |
. . . 4
⊢ (𝜑 → (𝑆 ∘ 𝑠) ∈ ran 𝐿) |
32 | 2, 3 | msubf 33237 |
. . . . . . . . 9
⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸) |
33 | 29, 32 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆:𝐸⟶𝐸) |
34 | | fco 6590 |
. . . . . . . 8
⊢ ((𝑆:𝐸⟶𝐸 ∧ 𝑠:𝐸⟶𝐸) → (𝑆 ∘ 𝑠):𝐸⟶𝐸) |
35 | 33, 5, 34 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (𝑆 ∘ 𝑠):𝐸⟶𝐸) |
36 | 35 | ffnd 6567 |
. . . . . 6
⊢ (𝜑 → (𝑆 ∘ 𝑠) Fn 𝐸) |
37 | 36 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → (𝑆 ∘ 𝑠) Fn 𝐸) |
38 | | mclsppslem.11 |
. . . . . . . . 9
⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) |
39 | 5 | ffund 6570 |
. . . . . . . . . 10
⊢ (𝜑 → Fun 𝑠) |
40 | 17 | simp2bi 1148 |
. . . . . . . . . . . . . 14
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin)) |
41 | 15, 40 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin)) |
42 | 41 | simpld 498 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑜 ⊆ 𝐸) |
43 | 26, 3, 27 | mvhf 33263 |
. . . . . . . . . . . . 13
⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
44 | | frn 6573 |
. . . . . . . . . . . . 13
⊢ (𝐻:𝑉⟶𝐸 → ran 𝐻 ⊆ 𝐸) |
45 | 6, 43, 44 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran 𝐻 ⊆ 𝐸) |
46 | 42, 45 | unssd 4116 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ 𝐸) |
47 | 5 | fdmd 6577 |
. . . . . . . . . . 11
⊢ (𝜑 → dom 𝑠 = 𝐸) |
48 | 46, 47 | sseqtrrd 3958 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠) |
49 | | funimass3 6895 |
. . . . . . . . . 10
⊢ ((Fun
𝑠 ∧ (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠) → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))))) |
50 | 39, 48, 49 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))))) |
51 | 38, 50 | mpbid 235 |
. . . . . . . 8
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))) |
52 | | cnvco 5771 |
. . . . . . . . . 10
⊢ ◡(𝑆 ∘ 𝑠) = (◡𝑠 ∘ ◡𝑆) |
53 | 52 | imaeq1i 5943 |
. . . . . . . . 9
⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) |
54 | | imaco 6132 |
. . . . . . . . 9
⊢ ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))) |
55 | 53, 54 | eqtri 2767 |
. . . . . . . 8
⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))) |
56 | 51, 55 | sseqtrrdi 3968 |
. . . . . . 7
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
57 | 56 | unssad 4117 |
. . . . . 6
⊢ (𝜑 → 𝑜 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
58 | 57 | sselda 3917 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
59 | | elpreima 6899 |
. . . . . 6
⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → (𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ (𝑐 ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵)))) |
60 | 59 | simplbda 503 |
. . . . 5
⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵)) |
61 | 37, 58, 60 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵)) |
62 | 36 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑆 ∘ 𝑠) Fn 𝐸) |
63 | 56 | unssbd 4118 |
. . . . . . 7
⊢ (𝜑 → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
64 | 63 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
65 | | ffn 6566 |
. . . . . . . 8
⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉) |
66 | 6, 43, 65 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → 𝐻 Fn 𝑉) |
67 | | fnfvelrn 6922 |
. . . . . . 7
⊢ ((𝐻 Fn 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻) |
68 | 66, 67 | sylan 583 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻) |
69 | 64, 68 | sseldd 3918 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
70 | | elpreima 6899 |
. . . . . 6
⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → ((𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑡) ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵)))) |
71 | 70 | simplbda 503 |
. . . . 5
⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵)) |
72 | 62, 69, 71 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵)) |
73 | 5 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑠:𝐸⟶𝐸) |
74 | 6, 43 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐻:𝑉⟶𝐸) |
75 | 74 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝐻:𝑉⟶𝐸) |
76 | 18 | simp1d 1144 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚)) |
77 | 76 | simpld 498 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑚 ⊆ 𝐷) |
78 | 26, 16 | mdvval 33209 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) |
79 | | difss 4062 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑉 × 𝑉) ∖ I ) ⊆ (𝑉 × 𝑉) |
80 | 78, 79 | eqsstri 3951 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐷 ⊆ (𝑉 × 𝑉) |
81 | 77, 80 | sstrdi 3929 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑚 ⊆ (𝑉 × 𝑉)) |
82 | 81 | ssbrd 5112 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑐𝑚𝑑 → 𝑐(𝑉 × 𝑉)𝑑)) |
83 | 82 | imp 410 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐(𝑉 × 𝑉)𝑑) |
84 | | brxp 5615 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐(𝑉 × 𝑉)𝑑 ↔ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) |
85 | 83, 84 | sylib 221 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) |
86 | 85 | simpld 498 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐 ∈ 𝑉) |
87 | 75, 86 | ffvelrnd 6926 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑐) ∈ 𝐸) |
88 | | fvco3 6831 |
. . . . . . . . . . . . 13
⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑐) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐)))) |
89 | 73, 87, 88 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐)))) |
90 | 89 | fveq2d 6742 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐))))) |
91 | 6 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑇 ∈ mFS) |
92 | 29 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑆 ∈ ran 𝐿) |
93 | 73, 87 | ffvelrnd 6926 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑐)) ∈ 𝐸) |
94 | 2, 3, 28, 27 | msubvrs 33265 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ mFS ∧ 𝑆 ∈ ran 𝐿 ∧ (𝑠‘(𝐻‘𝑐)) ∈ 𝐸) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪
𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))) |
95 | 91, 92, 93, 94 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪
𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))) |
96 | 90, 95 | eqtrd 2779 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = ∪
𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))) |
97 | 96 | eleq2d 2825 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ 𝑎 ∈ ∪
𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))) |
98 | | eliun 4924 |
. . . . . . . . 9
⊢ (𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))) |
99 | 97, 98 | bitrdi 290 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))))) |
100 | 85 | simprd 499 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑑 ∈ 𝑉) |
101 | 75, 100 | ffvelrnd 6926 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑑) ∈ 𝐸) |
102 | | fvco3 6831 |
. . . . . . . . . . . . 13
⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑑) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑)))) |
103 | 73, 101, 102 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑)))) |
104 | 103 | fveq2d 6742 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑))))) |
105 | 73, 101 | ffvelrnd 6926 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑑)) ∈ 𝐸) |
106 | 2, 3, 28, 27 | msubvrs 33265 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ mFS ∧ 𝑆 ∈ ran 𝐿 ∧ (𝑠‘(𝐻‘𝑑)) ∈ 𝐸) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪
𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))) |
107 | 91, 92, 105, 106 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪
𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))) |
108 | 104, 107 | eqtrd 2779 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = ∪
𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))) |
109 | 108 | eleq2d 2825 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ 𝑏 ∈ ∪
𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))) |
110 | | eliun 4924 |
. . . . . . . . 9
⊢ (𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) |
111 | 109, 110 | bitrdi 290 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) |
112 | 99, 111 | anbi12d 634 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) ↔ (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))) |
113 | | reeanv 3294 |
. . . . . . . 8
⊢
(∃𝑢 ∈
(𝑊‘(𝑠‘(𝐻‘𝑐)))∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) ↔ (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) |
114 | | simpll 767 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝜑) |
115 | | brxp 5615 |
. . . . . . . . . . . 12
⊢ (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 ↔ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) |
116 | | mclsppslem.12 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) |
117 | | breq12 5074 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧𝑚𝑤 ↔ 𝑐𝑚𝑑)) |
118 | | simpl 486 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐) |
119 | 118 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑧) = (𝐻‘𝑐)) |
120 | 119 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑧)) = (𝑠‘(𝐻‘𝑐))) |
121 | 120 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑧))) = (𝑊‘(𝑠‘(𝐻‘𝑐)))) |
122 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑) |
123 | 122 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑤) = (𝐻‘𝑑)) |
124 | 123 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑤)) = (𝑠‘(𝐻‘𝑑))) |
125 | 124 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑤))) = (𝑊‘(𝑠‘(𝐻‘𝑑)))) |
126 | 121, 125 | xpeq12d 5599 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) = ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))) |
127 | 126 | sseq1d 3948 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀 ↔ ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)) |
128 | 117, 127 | imbi12d 348 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) ↔ (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))) |
129 | 128 | spc2gv 3530 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑐 ∈ V ∧ 𝑑 ∈ V) → (∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀))) |
130 | 129 | el2v 3431 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)) |
131 | 116, 130 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)) |
132 | 131 | imp 410 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀) |
133 | 132 | ssbrd 5112 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 → 𝑢𝑀𝑣)) |
134 | 115, 133 | syl5bir 246 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))) → 𝑢𝑀𝑣)) |
135 | 134 | imp 410 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝑢𝑀𝑣) |
136 | | vex 3427 |
. . . . . . . . . . . . 13
⊢ 𝑢 ∈ V |
137 | | vex 3427 |
. . . . . . . . . . . . 13
⊢ 𝑣 ∈ V |
138 | | breq12 5074 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥𝑀𝑦 ↔ 𝑢𝑀𝑣)) |
139 | | simpl 486 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢) |
140 | 139 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑥) = (𝐻‘𝑢)) |
141 | 140 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑢))) |
142 | 141 | fveq2d 6742 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑢)))) |
143 | 142 | eleq2d 2825 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ↔ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))))) |
144 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣) |
145 | 144 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑦) = (𝐻‘𝑣)) |
146 | 145 | fveq2d 6742 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑣))) |
147 | 146 | fveq2d 6742 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑣)))) |
148 | 147 | eleq2d 2825 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) ↔ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) |
149 | 138, 143,
148 | 3anbi123d 1438 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦)))) ↔ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))))) |
150 | 149 | anbi2d 632 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) ↔ (𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))))) |
151 | 150 | imbi1d 345 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏) ↔ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏))) |
152 | | mclspps.8 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏) |
153 | 136, 137,
151, 152 | vtocl2 3491 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏) |
154 | 153 | 3exp2 1356 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑢𝑀𝑣 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))) → 𝑎𝐾𝑏)))) |
155 | 154 | imp4b 425 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢𝑀𝑣) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
156 | 114, 135,
155 | syl2anc 587 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
157 | 156 | rexlimdvva 3223 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
158 | 113, 157 | syl5bir 246 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
159 | 112, 158 | sylbid 243 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) → 𝑎𝐾𝑏)) |
160 | 159 | exp4b 434 |
. . . . 5
⊢ (𝜑 → (𝑐𝑚𝑑 → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) → 𝑎𝐾𝑏)))) |
161 | 160 | 3imp2 1351 |
. . . 4
⊢ ((𝜑 ∧ (𝑐𝑚𝑑 ∧ 𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))))) → 𝑎𝐾𝑏) |
162 | 16, 3, 23, 6, 24, 25, 7, 2, 26, 27, 28, 14, 31, 61, 72, 161 | mclsax 33274 |
. . 3
⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) ∈ (𝐾𝐶𝐵)) |
163 | 22, 162 | eqeltrrd 2841 |
. 2
⊢ (𝜑 → (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵)) |
164 | 33 | ffnd 6567 |
. . 3
⊢ (𝜑 → 𝑆 Fn 𝐸) |
165 | | elpreima 6899 |
. . 3
⊢ (𝑆 Fn 𝐸 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵)))) |
166 | 164, 165 | syl 17 |
. 2
⊢ (𝜑 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵)))) |
167 | 20, 163, 166 | mpbir2and 713 |
1
⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵))) |