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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 12087 | . . . . 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 28996 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑄 ∈ 𝑋) |
6 | 5 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑄 ∈ 𝑋) |
7 | 2, 6, 6 | 3jca 1127 | . . 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 29116 | . . 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 684 | . 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 29014 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑄 ∈ 𝑋) → ((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄) = 𝑄) |
17 | 5, 16 | mpdan 684 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → ((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄) = 𝑄) |
18 | 17 | fveq2d 6778 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = (𝑇‘𝑄)) |
19 | 18 | 3ad2ant1 1132 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = (𝑇‘𝑄)) |
20 | simp2 1136 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ NrmCVec) | |
21 | 3, 8, 13 | lnof 29117 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
22 | 21, 6 | ffvelrnd 6962 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) ∈ 𝑌) |
23 | lno0.z | . . . 4 ⊢ 𝑍 = (0vec‘𝑊) | |
24 | 8, 10, 12, 23 | nvlinv 29014 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑄) ∈ 𝑌) → ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝑄))( +𝑣 ‘𝑊)(𝑇‘𝑄)) = 𝑍) |
25 | 20, 22, 24 | syl2anc 584 | . 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 1086 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 1c1 10872 -cneg 11206 NrmCVeccnv 28946 +𝑣 cpv 28947 BaseSetcba 28948 ·𝑠OLD cns 28949 0veccn0v 28950 LnOp clno 29102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-sub 11207 df-neg 11208 df-grpo 28855 df-gid 28856 df-ginv 28857 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-nmcv 28962 df-lno 29106 |
This theorem is referenced by: lnomul 29122 nmlno0lem 29155 nmlnoubi 29158 lnon0 29160 nmblolbii 29161 blocnilem 29166 |
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