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 32028 |
. . . 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 2777 |
. . . . . . . . 9
⊢
(mAx‘𝑇) =
(mAx‘𝑇) |
8 | | eqid 2777 |
. . . . . . . . 9
⊢
(mStat‘𝑇) =
(mStat‘𝑇) |
9 | 7, 8 | maxsta 32050 |
. . . . . . . 8
⊢ (𝑇 ∈ mFS →
(mAx‘𝑇) ⊆
(mStat‘𝑇)) |
10 | 6, 9 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (mAx‘𝑇) ⊆ (mStat‘𝑇)) |
11 | | eqid 2777 |
. . . . . . . 8
⊢
(mPreSt‘𝑇) =
(mPreSt‘𝑇) |
12 | 11, 8 | mstapst 32043 |
. . . . . . 7
⊢
(mStat‘𝑇)
⊆ (mPreSt‘𝑇) |
13 | 10, 12 | syl6ss 3832 |
. . . . . 6
⊢ (𝜑 → (mAx‘𝑇) ⊆ (mPreSt‘𝑇)) |
14 | | mclsppslem.9 |
. . . . . 6
⊢ (𝜑 → 〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇)) |
15 | 13, 14 | sseldd 3821 |
. . . . 5
⊢ (𝜑 → 〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇)) |
16 | | mclspps.d |
. . . . . 6
⊢ 𝐷 = (mDV‘𝑇) |
17 | 16, 3, 11 | elmpst 32032 |
. . . . 5
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸)) |
18 | 15, 17 | sylib 210 |
. . . 4
⊢ (𝜑 → ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸)) |
19 | 18 | simp3d 1135 |
. . 3
⊢ (𝜑 → 𝑝 ∈ 𝐸) |
20 | 5, 19 | ffvelrnd 6624 |
. 2
⊢ (𝜑 → (𝑠‘𝑝) ∈ 𝐸) |
21 | | fvco3 6535 |
. . . 4
⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘𝑝) = (𝑆‘(𝑠‘𝑝))) |
22 | 5, 19, 21 | syl2anc 579 |
. . 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 32027 |
. . . . 5
⊢ ((𝑆 ∈ ran 𝐿 ∧ 𝑠 ∈ ran 𝐿) → (𝑆 ∘ 𝑠) ∈ ran 𝐿) |
31 | 29, 1, 30 | syl2anc 579 |
. . . 4
⊢ (𝜑 → (𝑆 ∘ 𝑠) ∈ ran 𝐿) |
32 | 2, 3 | msubf 32028 |
. . . . . . . . 9
⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸) |
33 | 29, 32 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆:𝐸⟶𝐸) |
34 | | fco 6308 |
. . . . . . . 8
⊢ ((𝑆:𝐸⟶𝐸 ∧ 𝑠:𝐸⟶𝐸) → (𝑆 ∘ 𝑠):𝐸⟶𝐸) |
35 | 33, 5, 34 | syl2anc 579 |
. . . . . . 7
⊢ (𝜑 → (𝑆 ∘ 𝑠):𝐸⟶𝐸) |
36 | 35 | ffnd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑆 ∘ 𝑠) Fn 𝐸) |
37 | 36 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → (𝑆 ∘ 𝑠) Fn 𝐸) |
38 | | mclsppslem.11 |
. . . . . . . . 9
⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) |
39 | 5 | ffund 6295 |
. . . . . . . . . 10
⊢ (𝜑 → Fun 𝑠) |
40 | 17 | simp2bi 1137 |
. . . . . . . . . . . . . 14
⊢
(〈𝑚, 𝑜, 𝑝〉 ∈ (mPreSt‘𝑇) → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin)) |
41 | 15, 40 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin)) |
42 | 41 | simpld 490 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑜 ⊆ 𝐸) |
43 | 26, 3, 27 | mvhf 32054 |
. . . . . . . . . . . . 13
⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
44 | | frn 6297 |
. . . . . . . . . . . . 13
⊢ (𝐻:𝑉⟶𝐸 → ran 𝐻 ⊆ 𝐸) |
45 | 6, 43, 44 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran 𝐻 ⊆ 𝐸) |
46 | 42, 45 | unssd 4011 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ 𝐸) |
47 | 5 | fdmd 6300 |
. . . . . . . . . . 11
⊢ (𝜑 → dom 𝑠 = 𝐸) |
48 | 46, 47 | sseqtr4d 3860 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ dom 𝑠) |
49 | | funimass3 6596 |
. . . . . . . . . 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 579 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))))) |
51 | 38, 50 | mpbid 224 |
. . . . . . . 8
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵)))) |
52 | | cnvco 5553 |
. . . . . . . . . 10
⊢ ◡(𝑆 ∘ 𝑠) = (◡𝑠 ∘ ◡𝑆) |
53 | 52 | imaeq1i 5717 |
. . . . . . . . 9
⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) |
54 | | imaco 5894 |
. . . . . . . . 9
⊢ ((◡𝑠 ∘ ◡𝑆) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))) |
55 | 53, 54 | eqtri 2801 |
. . . . . . . 8
⊢ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) = (◡𝑠 “ (◡𝑆 “ (𝐾𝐶𝐵))) |
56 | 51, 55 | syl6sseqr 3870 |
. . . . . . 7
⊢ (𝜑 → (𝑜 ∪ ran 𝐻) ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
57 | 56 | unssad 4012 |
. . . . . 6
⊢ (𝜑 → 𝑜 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
58 | 57 | sselda 3820 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
59 | | elpreima 6600 |
. . . . . 6
⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → (𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ (𝑐 ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵)))) |
60 | 59 | simplbda 495 |
. . . . 5
⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ 𝑐 ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵)) |
61 | 37, 58, 60 | syl2anc 579 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ 𝑜) → ((𝑆 ∘ 𝑠)‘𝑐) ∈ (𝐾𝐶𝐵)) |
62 | 36 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑆 ∘ 𝑠) Fn 𝐸) |
63 | 56 | unssbd 4013 |
. . . . . . 7
⊢ (𝜑 → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
64 | 63 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ran 𝐻 ⊆ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
65 | | ffn 6291 |
. . . . . . . 8
⊢ (𝐻:𝑉⟶𝐸 → 𝐻 Fn 𝑉) |
66 | 6, 43, 65 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → 𝐻 Fn 𝑉) |
67 | | fnfvelrn 6620 |
. . . . . . 7
⊢ ((𝐻 Fn 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻) |
68 | 66, 67 | sylan 575 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ ran 𝐻) |
69 | 64, 68 | sseldd 3821 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) |
70 | | elpreima 6600 |
. . . . . 6
⊢ ((𝑆 ∘ 𝑠) Fn 𝐸 → ((𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑡) ∈ 𝐸 ∧ ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵)))) |
71 | 70 | simplbda 495 |
. . . . 5
⊢ (((𝑆 ∘ 𝑠) Fn 𝐸 ∧ (𝐻‘𝑡) ∈ (◡(𝑆 ∘ 𝑠) “ (𝐾𝐶𝐵))) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵)) |
72 | 62, 69, 71 | syl2anc 579 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑡)) ∈ (𝐾𝐶𝐵)) |
73 | 5 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑠:𝐸⟶𝐸) |
74 | 6, 43 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐻:𝑉⟶𝐸) |
75 | 74 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝐻:𝑉⟶𝐸) |
76 | 18 | simp1d 1133 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚)) |
77 | 76 | simpld 490 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑚 ⊆ 𝐷) |
78 | 26, 16 | mdvval 32000 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) |
79 | | difss 3959 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑉 × 𝑉) ∖ I ) ⊆ (𝑉 × 𝑉) |
80 | 78, 79 | eqsstri 3853 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐷 ⊆ (𝑉 × 𝑉) |
81 | 77, 80 | syl6ss 3832 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑚 ⊆ (𝑉 × 𝑉)) |
82 | 81 | ssbrd 4929 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑐𝑚𝑑 → 𝑐(𝑉 × 𝑉)𝑑)) |
83 | 82 | imp 397 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐(𝑉 × 𝑉)𝑑) |
84 | | brxp 5401 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐(𝑉 × 𝑉)𝑑 ↔ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) |
85 | 83, 84 | sylib 210 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) |
86 | 85 | simpld 490 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑐 ∈ 𝑉) |
87 | 75, 86 | ffvelrnd 6624 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑐) ∈ 𝐸) |
88 | | fvco3 6535 |
. . . . . . . . . . . . 13
⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑐) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐)))) |
89 | 73, 87, 88 | syl2anc 579 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑐)) = (𝑆‘(𝑠‘(𝐻‘𝑐)))) |
90 | 89 | fveq2d 6450 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐))))) |
91 | 6 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑇 ∈ mFS) |
92 | 29 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑆 ∈ ran 𝐿) |
93 | 73, 87 | ffvelrnd 6624 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑐)) ∈ 𝐸) |
94 | 2, 3, 28, 27 | msubvrs 32056 |
. . . . . . . . . . . 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 1439 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑐)))) = ∪
𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))) |
96 | 90, 95 | eqtrd 2813 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) = ∪
𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢)))) |
97 | 96 | eleq2d 2844 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ 𝑎 ∈ ∪
𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))))) |
98 | | eliun 4757 |
. . . . . . . . 9
⊢ (𝑎 ∈ ∪ 𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))(𝑊‘(𝑆‘(𝐻‘𝑢))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢)))) |
99 | 97, 98 | syl6bb 279 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ↔ ∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))))) |
100 | 85 | simprd 491 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → 𝑑 ∈ 𝑉) |
101 | 75, 100 | ffvelrnd 6624 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝐻‘𝑑) ∈ 𝐸) |
102 | | fvco3 6535 |
. . . . . . . . . . . . 13
⊢ ((𝑠:𝐸⟶𝐸 ∧ (𝐻‘𝑑) ∈ 𝐸) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑)))) |
103 | 73, 101, 102 | syl2anc 579 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)) = (𝑆‘(𝑠‘(𝐻‘𝑑)))) |
104 | 103 | fveq2d 6450 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑))))) |
105 | 73, 101 | ffvelrnd 6624 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑠‘(𝐻‘𝑑)) ∈ 𝐸) |
106 | 2, 3, 28, 27 | msubvrs 32056 |
. . . . . . . . . . . 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 1439 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘(𝑆‘(𝑠‘(𝐻‘𝑑)))) = ∪
𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))) |
108 | 104, 107 | eqtrd 2813 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) = ∪
𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣)))) |
109 | 108 | eleq2d 2844 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ 𝑏 ∈ ∪
𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))))) |
110 | | eliun 4757 |
. . . . . . . . 9
⊢ (𝑏 ∈ ∪ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑊‘(𝑆‘(𝐻‘𝑣))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) |
111 | 109, 110 | syl6bb 279 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) ↔ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) |
112 | 99, 111 | anbi12d 624 |
. . . . . . 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 3292 |
. . . . . . . 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 757 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝜑) |
115 | | brxp 5401 |
. . . . . . . . . . . 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 | | vex 3400 |
. . . . . . . . . . . . . . . 16
⊢ 𝑐 ∈ V |
118 | | vex 3400 |
. . . . . . . . . . . . . . . 16
⊢ 𝑑 ∈ V |
119 | | breq12 4891 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧𝑚𝑤 ↔ 𝑐𝑚𝑑)) |
120 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐) |
121 | 120 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑧) = (𝐻‘𝑐)) |
122 | 121 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑧)) = (𝑠‘(𝐻‘𝑐))) |
123 | 122 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑧))) = (𝑊‘(𝑠‘(𝐻‘𝑐)))) |
124 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑) |
125 | 124 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝐻‘𝑤) = (𝐻‘𝑑)) |
126 | 125 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑠‘(𝐻‘𝑤)) = (𝑠‘(𝐻‘𝑑))) |
127 | 126 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑊‘(𝑠‘(𝐻‘𝑤))) = (𝑊‘(𝑠‘(𝐻‘𝑑)))) |
128 | 123, 127 | xpeq12d 5386 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) = ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))) |
129 | 128 | sseq1d 3850 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀 ↔ ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)) |
130 | 119, 129 | imbi12d 336 |
. . . . . . . . . . . . . . . . 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))) |
131 | 130 | spc2gv 3497 |
. . . . . . . . . . . . . . . 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))) |
132 | 117, 118,
131 | mp2an 682 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀) → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)) |
133 | 116, 132 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑐𝑚𝑑 → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀)) |
134 | 133 | imp 397 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑)))) ⊆ 𝑀) |
135 | 134 | ssbrd 4929 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (𝑢((𝑊‘(𝑠‘(𝐻‘𝑐))) × (𝑊‘(𝑠‘(𝐻‘𝑑))))𝑣 → 𝑢𝑀𝑣)) |
136 | 115, 135 | syl5bir 235 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))) → 𝑢𝑀𝑣)) |
137 | 136 | imp 397 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → 𝑢𝑀𝑣) |
138 | | vex 3400 |
. . . . . . . . . . . . 13
⊢ 𝑢 ∈ V |
139 | | vex 3400 |
. . . . . . . . . . . . 13
⊢ 𝑣 ∈ V |
140 | | breq12 4891 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥𝑀𝑦 ↔ 𝑢𝑀𝑣)) |
141 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢) |
142 | 141 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑥) = (𝐻‘𝑢)) |
143 | 142 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑥)) = (𝑆‘(𝐻‘𝑢))) |
144 | 143 | fveq2d 6450 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑥))) = (𝑊‘(𝑆‘(𝐻‘𝑢)))) |
145 | 144 | eleq2d 2844 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ↔ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))))) |
146 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣) |
147 | 146 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝐻‘𝑦) = (𝐻‘𝑣)) |
148 | 147 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑆‘(𝐻‘𝑦)) = (𝑆‘(𝐻‘𝑣))) |
149 | 148 | fveq2d 6450 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑊‘(𝑆‘(𝐻‘𝑦))) = (𝑊‘(𝑆‘(𝐻‘𝑣)))) |
150 | 149 | eleq2d 2844 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))) ↔ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) |
151 | 140, 145,
150 | 3anbi123d 1509 |
. . . . . . . . . . . . . . 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)))))) |
152 | 151 | anbi2d 622 |
. . . . . . . . . . . . . 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))))))) |
153 | 152 | imbi1d 333 |
. . . . . . . . . . . . 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))) |
154 | | mclspps.8 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏) |
155 | 138, 139,
153, 154 | vtocl2 3461 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢𝑀𝑣 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))))) → 𝑎𝐾𝑏) |
156 | 155 | 3exp2 1416 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑢𝑀𝑣 → (𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) → (𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣))) → 𝑎𝐾𝑏)))) |
157 | 156 | imp4b 414 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢𝑀𝑣) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
158 | 114, 137,
157 | syl2anc 579 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑐𝑚𝑑) ∧ (𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐))) ∧ 𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑))))) → ((𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
159 | 158 | rexlimdvva 3220 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → (∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))(𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
160 | 113, 159 | syl5bir 235 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((∃𝑢 ∈ (𝑊‘(𝑠‘(𝐻‘𝑐)))𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑢))) ∧ ∃𝑣 ∈ (𝑊‘(𝑠‘(𝐻‘𝑑)))𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑣)))) → 𝑎𝐾𝑏)) |
161 | 112, 160 | sylbid 232 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐𝑚𝑑) → ((𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑)))) → 𝑎𝐾𝑏)) |
162 | 161 | exp4b 423 |
. . . . 5
⊢ (𝜑 → (𝑐𝑚𝑑 → (𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) → (𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))) → 𝑎𝐾𝑏)))) |
163 | 162 | 3imp2 1411 |
. . . 4
⊢ ((𝜑 ∧ (𝑐𝑚𝑑 ∧ 𝑎 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑐))) ∧ 𝑏 ∈ (𝑊‘((𝑆 ∘ 𝑠)‘(𝐻‘𝑑))))) → 𝑎𝐾𝑏) |
164 | 16, 3, 23, 6, 24, 25, 7, 2, 26, 27, 28, 14, 31, 61, 72, 163 | mclsax 32065 |
. . 3
⊢ (𝜑 → ((𝑆 ∘ 𝑠)‘𝑝) ∈ (𝐾𝐶𝐵)) |
165 | 22, 164 | eqeltrrd 2859 |
. 2
⊢ (𝜑 → (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵)) |
166 | 33 | ffnd 6292 |
. . 3
⊢ (𝜑 → 𝑆 Fn 𝐸) |
167 | | elpreima 6600 |
. . 3
⊢ (𝑆 Fn 𝐸 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵)))) |
168 | 166, 167 | syl 17 |
. 2
⊢ (𝜑 → ((𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝑠‘𝑝) ∈ 𝐸 ∧ (𝑆‘(𝑠‘𝑝)) ∈ (𝐾𝐶𝐵)))) |
169 | 20, 165, 168 | mpbir2and 703 |
1
⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵))) |