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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > resvlem | Structured version Visualization version GIF version |
Description: Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
Ref | Expression |
---|---|
resvlem.r | ⊢ 𝑅 = (𝑊 ↾v 𝐴) |
resvlem.e | ⊢ 𝐶 = (𝐸‘𝑊) |
resvlem.f | ⊢ 𝐸 = Slot (𝐸‘ndx) |
resvlem.n | ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
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 2735 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2735 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
5 | 2, 3, 4 | resvid2 33334 | . . . . . 6 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
6 | 5 | fveq2d 6911 | . . . . 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 33335 | . . . . . . 7 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
9 | 8 | fveq2d 6911 | . . . . . 6 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉))) |
10 | resvlem.f | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
11 | resvlem.n | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) | |
12 | 10, 11 | setsnid 17243 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
13 | 9, 12 | eqtr4di 2793 | . . . . 5 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
14 | 13 | 3expib 1121 | . . . 4 ⊢ (¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
15 | 7, 14 | pm2.61i 182 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
16 | 10 | str0 17223 | . . . . . . 7 ⊢ ∅ = (𝐸‘∅) |
17 | 16 | eqcomi 2744 | . . . . . 6 ⊢ (𝐸‘∅) = ∅ |
18 | reldmresv 33332 | . . . . . 6 ⊢ Rel dom ↾v | |
19 | 17, 2, 18 | oveqprc 17226 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘𝑅)) |
20 | 19 | eqcomd 2741 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
21 | 20 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
22 | 15, 21 | pm2.61ian 812 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
23 | 1, 22 | eqtr4id 2794 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 〈cop 4637 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 Slot cslot 17215 ndxcnx 17227 Basecbs 17245 ↾s cress 17274 Scalarcsca 17301 ↾v cresv 33330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17198 df-slot 17216 df-resv 33331 |
This theorem is referenced by: resvbas 33339 resvplusg 33341 resvvsca 33343 resvmulr 33345 |
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