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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 12378 | . . . . 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 30663 | . . . . 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 2735 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
10 | eqid 2735 | . . . 4 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
11 | eqid 2735 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
12 | eqid 2735 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
13 | lno0.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
14 | 3, 8, 9, 10, 11, 12, 13 | lnolin 30783 | . . 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 30681 | . . . . 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 6911 | . . 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 30784 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
22 | 21, 6 | ffvelcdmd 7105 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) ∈ 𝑌) |
23 | lno0.z | . . . 4 ⊢ 𝑍 = (0vec‘𝑊) | |
24 | 8, 10, 12, 23 | nvlinv 30681 | . . 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 2783 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 -cneg 11491 NrmCVeccnv 30613 +𝑣 cpv 30614 BaseSetcba 30615 ·𝑠OLD cns 30616 0veccn0v 30617 LnOp clno 30769 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-grpo 30522 df-gid 30523 df-ginv 30524 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-nmcv 30629 df-lno 30773 |
This theorem is referenced by: lnomul 30789 nmlno0lem 30822 nmlnoubi 30825 lnon0 30827 nmblolbii 30828 blocnilem 30833 |
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