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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 2736 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 4 | eqid 2736 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 5 | 2, 3, 4 | resvid2 33355 | . . . . . 6 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) | 
| 6 | 5 | fveq2d 6909 | . . . . 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 33356 | . . . . . . 7 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) | 
| 9 | 8 | fveq2d 6909 | . . . . . 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 17246 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) | 
| 13 | 9, 12 | eqtr4di 2794 | . . . . 5 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) | 
| 14 | 13 | 3expib 1122 | . . . 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 17227 | . . . . . . 7 ⊢ ∅ = (𝐸‘∅) | 
| 17 | 16 | eqcomi 2745 | . . . . . 6 ⊢ (𝐸‘∅) = ∅ | 
| 18 | reldmresv 33353 | . . . . . 6 ⊢ Rel dom ↾v | |
| 19 | 17, 2, 18 | oveqprc 17230 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = (𝐸‘𝑅)) | 
| 20 | 19 | eqcomd 2742 | . . . 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 811 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) | 
| 23 | 1, 22 | eqtr4id 2795 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 〈cop 4631 ‘cfv 6560 (class class class)co 7432 sSet csts 17201 Slot cslot 17219 ndxcnx 17231 Basecbs 17248 ↾s cress 17275 Scalarcsca 17301 ↾v cresv 33351 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-sets 17202 df-slot 17220 df-resv 33352 | 
| This theorem is referenced by: resvbas 33360 resvplusg 33362 resvvsca 33364 resvmulr 33366 | 
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