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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.r | . . . . . . 7 ⊢ 𝑅 = (𝑊 ↾v 𝐴) | |
2 | eqid 2610 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2610 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
4 | 1, 2, 3 | resvid2 29159 | . . . . . 6 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = 𝑊) |
5 | 4 | fveq2d 6107 | . . . . 5 ⊢ (((Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
6 | 5 | 3expib 1260 | . . . 4 ⊢ ((Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
7 | 1, 2, 3 | resvval2 29160 | . . . . . . 7 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
8 | 7 | fveq2d 6107 | . . . . . 6 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉))) |
9 | resvlem.f | . . . . . . . 8 ⊢ 𝐸 = Slot 𝑁 | |
10 | resvlem.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
11 | 9, 10 | ndxid 15716 | . . . . . . 7 ⊢ 𝐸 = Slot (𝐸‘ndx) |
12 | 9, 10 | ndxarg 15715 | . . . . . . . . 9 ⊢ (𝐸‘ndx) = 𝑁 |
13 | resvlem.b | . . . . . . . . 9 ⊢ 𝑁 ≠ 5 | |
14 | 12, 13 | eqnetri 2852 | . . . . . . . 8 ⊢ (𝐸‘ndx) ≠ 5 |
15 | scandx 15836 | . . . . . . . 8 ⊢ (Scalar‘ndx) = 5 | |
16 | 14, 15 | neeqtrri 2855 | . . . . . . 7 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
17 | 11, 16 | setsnid 15743 | . . . . . 6 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), ((Scalar‘𝑊) ↾s 𝐴)〉)) |
18 | 8, 17 | syl6eqr 2662 | . . . . 5 ⊢ ((¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
19 | 18 | 3expib 1260 | . . . 4 ⊢ (¬ (Base‘(Scalar‘𝑊)) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊))) |
20 | 6, 19 | pm2.61i 175 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
21 | reldmresv 29157 | . . . . . . . . 9 ⊢ Rel dom ↾v | |
22 | 21 | ovprc1 6582 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾v 𝐴) = ∅) |
23 | 1, 22 | syl5eq 2656 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → 𝑅 = ∅) |
24 | 23 | fveq2d 6107 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘∅)) |
25 | 9 | str0 15739 | . . . . . 6 ⊢ ∅ = (𝐸‘∅) |
26 | 24, 25 | syl6eqr 2662 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = ∅) |
27 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
28 | 26, 27 | eqtr4d 2647 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑅) = (𝐸‘𝑊)) |
29 | 28 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐸‘𝑅) = (𝐸‘𝑊)) |
30 | 20, 29 | pm2.61ian 827 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝑅) = (𝐸‘𝑊)) |
31 | resvlem.e | . 2 ⊢ 𝐶 = (𝐸‘𝑊) | |
32 | 30, 31 | syl6reqr 2663 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 〈cop 4131 ‘cfv 5804 (class class class)co 6549 ℕcn 10897 5c5 10950 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 ↾v cresv 29155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-ndx 15698 df-slot 15699 df-sets 15701 df-sca 15784 df-resv 29156 |
This theorem is referenced by: resvbas 29163 resvplusg 29164 resvvsca 29165 resvmulr 29166 |
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