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Mirrors > Home > MPE Home > Th. List > Mathboxes > resvlemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of resvlem 31767 as of 31-Oct-2024. Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
resvlemOLD.r | ⊢ 𝑅 = (𝑊 ↾v 𝐴) |
resvlemOLD.e | ⊢ 𝐶 = (𝐸‘𝑊) |
resvlemOLD.f | ⊢ 𝐸 = Slot 𝑁 |
resvlemOLD.n | ⊢ 𝑁 ∈ ℕ |
resvlemOLD.b | ⊢ 𝑁 ≠ 5 |
Ref | Expression |
---|---|
resvlemOLD | ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resvlemOLD.e | . 2 ⊢ 𝐶 = (𝐸‘𝑊) | |
2 | resvlemOLD.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾v 𝐴) | |
3 | eqid 2736 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2736 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
5 | 2, 3, 4 | resvid2 31764 | . . . . . 6 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
6 | 5 | fveq2d 6823 | . . . . 5 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
7 | 6 | 3expib 1121 | . . . 4 ⊢ ((Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
8 | 2, 3, 4 | resvval2 31765 | . . . . . . 7 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
9 | 8 | fveq2d 6823 | . . . . . 6 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉))) |
10 | resvlemOLD.f | . . . . . . . 8 ⊢ 𝐸 = Slot 𝑁 | |
11 | resvlemOLD.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
12 | 10, 11 | ndxid 16987 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
13 | 10, 11 | ndxarg 16986 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
14 | resvlemOLD.b | . . . . . . . . 9 ⊢ 𝑁 ≠ 5 | |
15 | 13, 14 | eqnetri 3011 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 5 |
16 | scandx 17113 | . . . . . . . 8 ⊢ (Scalar‘ndx) = 5 | |
17 | 15, 16 | neeqtrri 3014 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
18 | 12, 17 | setsnid 16999 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
19 | 9, 18 | eqtr4di 2794 | . . . . 5 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
20 | 19 | 3expib 1121 | . . . 4 ⊢ (¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
21 | 7, 20 | pm2.61i 182 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
22 | reldmresv 31762 | . . . . . . . . 9 ⊢ Rel dom ↾v | |
23 | 22 | ovprc1 7368 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾v 𝐴) = ∅) |
24 | 2, 23 | eqtrid 2788 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
25 | 24 | fveq2d 6823 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
26 | 10 | str0 16979 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
27 | 25, 26 | eqtr4di 2794 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
28 | fvprc 6811 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
29 | 27, 28 | eqtr4d 2779 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
30 | 29 | adantr 481 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
31 | 21, 30 | pm2.61ian 809 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
32 | 1, 31 | eqtr4id 2795 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 Vcvv 3441 ⊆ wss 3897 ∅c0 4268 〈cop 4578 ‘cfv 6473 (class class class)co 7329 ℕcn 12066 5c5 12124 sSet csts 16953 Slot cslot 16971 ndxcnx 16983 Basecbs 17001 ↾s cress 17030 Scalarcsca 17054 ↾v cresv 31760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-1cn 11022 ax-addcl 11024 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-sets 16954 df-slot 16972 df-ndx 16984 df-sca 17067 df-resv 31761 |
This theorem is referenced by: resvbasOLD 31770 resvplusgOLD 31772 resvvscaOLD 31774 resvmulrOLD 31776 |
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