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Mirrors > Home > MPE Home > Th. List > Mathboxes > resvlemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of resvlem 32122 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 32119 | . . . . . 6 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
6 | 5 | fveq2d 6846 | . . . . 5 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
7 | 6 | 3expib 1122 | . . . 4 ⊢ ((Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
8 | 2, 3, 4 | resvval2 32120 | . . . . . . 7 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
9 | 8 | fveq2d 6846 | . . . . . 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 17069 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
13 | 10, 11 | ndxarg 17068 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
14 | resvlemOLD.b | . . . . . . . . 9 ⊢ 𝑁 ≠ 5 | |
15 | 13, 14 | eqnetri 3014 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 5 |
16 | scandx 17195 | . . . . . . . 8 ⊢ (Scalar‘ndx) = 5 | |
17 | 15, 16 | neeqtrri 3017 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
18 | 12, 17 | setsnid 17081 | . . . . . 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 1122 | . . . 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 32117 | . . . . . . . . 9 ⊢ Rel dom ↾v | |
23 | 22 | ovprc1 7396 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾v 𝐴) = ∅) |
24 | 2, 23 | eqtrid 2788 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
25 | 24 | fveq2d 6846 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
26 | 10 | str0 17061 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
27 | 25, 26 | eqtr4di 2794 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
28 | fvprc 6834 | . . . . 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 810 | . 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 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 〈cop 4592 ‘cfv 6496 (class class class)co 7357 ℕcn 12153 5c5 12211 sSet csts 17035 Slot cslot 17053 ndxcnx 17065 Basecbs 17083 ↾s cress 17112 Scalarcsca 17136 ↾v cresv 32115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-1cn 11109 ax-addcl 11111 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-sets 17036 df-slot 17054 df-ndx 17066 df-sca 17149 df-resv 32116 |
This theorem is referenced by: resvbasOLD 32125 resvplusgOLD 32127 resvvscaOLD 32129 resvmulrOLD 32131 |
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