![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > resvlem | Structured version Visualization version GIF version |
Description: Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
Ref | Expression |
---|---|
resvlem.r | ⊢ 𝑅 = (𝑊 ↾v 𝐴) |
resvlem.e | ⊢ 𝐶 = (𝐸‘𝑊) |
resvlem.f | ⊢ 𝐸 = Slot 𝑁 |
resvlem.n | ⊢ 𝑁 ∈ ℕ |
resvlem.b | ⊢ 𝑁 ≠ 5 |
Ref | Expression |
---|---|
resvlem | ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resvlem.e | . 2 ⊢ 𝐶 = (𝐸‘𝑊) | |
2 | resvlem.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾v 𝐴) | |
3 | eqid 2798 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2798 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
5 | 2, 3, 4 | resvid2 30952 | . . . . . 6 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
6 | 5 | fveq2d 6649 | . . . . 5 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
7 | 6 | 3expib 1119 | . . . 4 ⊢ ((Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
8 | 2, 3, 4 | resvval2 30953 | . . . . . . 7 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
9 | 8 | fveq2d 6649 | . . . . . 6 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉))) |
10 | resvlem.f | . . . . . . . 8 ⊢ 𝐸 = Slot 𝑁 | |
11 | resvlem.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
12 | 10, 11 | ndxid 16501 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
13 | 10, 11 | ndxarg 16500 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
14 | resvlem.b | . . . . . . . . 9 ⊢ 𝑁 ≠ 5 | |
15 | 13, 14 | eqnetri 3057 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 5 |
16 | scandx 16624 | . . . . . . . 8 ⊢ (Scalar‘ndx) = 5 | |
17 | 15, 16 | neeqtrri 3060 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
18 | 12, 17 | setsnid 16531 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
19 | 9, 18 | eqtr4di 2851 | . . . . 5 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
20 | 19 | 3expib 1119 | . . . 4 ⊢ (¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
21 | 7, 20 | pm2.61i 185 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
22 | reldmresv 30950 | . . . . . . . . 9 ⊢ Rel dom ↾v | |
23 | 22 | ovprc1 7174 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾v 𝐴) = ∅) |
24 | 2, 23 | syl5eq 2845 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
25 | 24 | fveq2d 6649 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
26 | 10 | str0 16527 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
27 | 25, 26 | eqtr4di 2851 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
28 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
29 | 27, 28 | eqtr4d 2836 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
30 | 29 | adantr 484 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
31 | 21, 30 | pm2.61ian 811 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
32 | 1, 31 | eqtr4id 2852 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ℕcn 11625 5c5 11683 ndxcnx 16472 sSet csts 16473 Slot cslot 16474 Basecbs 16475 ↾s cress 16476 Scalarcsca 16560 ↾v cresv 30948 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-ndx 16478 df-slot 16479 df-sets 16482 df-sca 16573 df-resv 30949 |
This theorem is referenced by: resvbas 30956 resvplusg 30957 resvvsca 30958 resvmulr 30959 |
Copyright terms: Public domain | W3C validator |