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Mirrors > Home > MPE Home > Th. List > lno0 | Structured version Visualization version GIF version |
Description: The value of a linear operator at zero is zero. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lno0.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
lno0.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
lno0.5 | ⊢ 𝑄 = (0vec‘𝑈) |
lno0.z | ⊢ 𝑍 = (0vec‘𝑊) |
lno0.7 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
Ref | Expression |
---|---|
lno0 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 11968 | . . . . 5 ⊢ -1 ∈ ℂ | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → -1 ∈ ℂ) |
3 | lno0.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | lno0.5 | . . . . . 6 ⊢ 𝑄 = (0vec‘𝑈) | |
5 | 3, 4 | nvzcl 28739 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑄 ∈ 𝑋) |
6 | 5 | 3ad2ant1 1135 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑄 ∈ 𝑋) |
7 | 2, 6, 6 | 3jca 1130 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (-1 ∈ ℂ ∧ 𝑄 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋)) |
8 | lno0.2 | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
9 | eqid 2738 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
10 | eqid 2738 | . . . 4 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
11 | eqid 2738 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
12 | eqid 2738 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
13 | lno0.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
14 | 3, 8, 9, 10, 11, 12, 13 | lnolin 28859 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (-1 ∈ ℂ ∧ 𝑄 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋)) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝑄))( +𝑣 ‘𝑊)(𝑇‘𝑄))) |
15 | 7, 14 | mpdan 687 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝑄))( +𝑣 ‘𝑊)(𝑇‘𝑄))) |
16 | 3, 9, 11, 4 | nvlinv 28757 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑄 ∈ 𝑋) → ((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄) = 𝑄) |
17 | 5, 16 | mpdan 687 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → ((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄) = 𝑄) |
18 | 17 | fveq2d 6739 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = (𝑇‘𝑄)) |
19 | 18 | 3ad2ant1 1135 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = (𝑇‘𝑄)) |
20 | simp2 1139 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ NrmCVec) | |
21 | 3, 8, 13 | lnof 28860 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
22 | 21, 6 | ffvelrnd 6923 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) ∈ 𝑌) |
23 | lno0.z | . . . 4 ⊢ 𝑍 = (0vec‘𝑊) | |
24 | 8, 10, 12, 23 | nvlinv 28757 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑄) ∈ 𝑌) → ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝑄))( +𝑣 ‘𝑊)(𝑇‘𝑄)) = 𝑍) |
25 | 20, 22, 24 | syl2anc 587 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝑄))( +𝑣 ‘𝑊)(𝑇‘𝑄)) = 𝑍) |
26 | 15, 19, 25 | 3eqtr3d 2786 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ‘cfv 6397 (class class class)co 7231 ℂcc 10751 1c1 10754 -cneg 11087 NrmCVeccnv 28689 +𝑣 cpv 28690 BaseSetcba 28691 ·𝑠OLD cns 28692 0veccn0v 28693 LnOp clno 28845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-po 5482 df-so 5483 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-1st 7779 df-2nd 7780 df-er 8411 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-ltxr 10896 df-sub 11088 df-neg 11089 df-grpo 28598 df-gid 28599 df-ginv 28600 df-ablo 28650 df-vc 28664 df-nv 28697 df-va 28700 df-ba 28701 df-sm 28702 df-0v 28703 df-nmcv 28705 df-lno 28849 |
This theorem is referenced by: lnomul 28865 nmlno0lem 28898 nmlnoubi 28901 lnon0 28903 nmblolbii 28904 blocnilem 28909 |
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