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Mirrors > Home > MPE Home > Th. List > lnosub | Structured version Visualization version GIF version |
Description: Subtraction property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnosub.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
lnosub.5 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
lnosub.6 | ⊢ 𝑁 = ( −𝑣 ‘𝑊) |
lnosub.7 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
Ref | Expression |
---|---|
lnosub | ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑀𝐵)) = ((𝑇‘𝐴)𝑁(𝑇‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 12378 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | lnosub.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | eqid 2735 | . . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
4 | eqid 2735 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
5 | eqid 2735 | . . . . 5 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
6 | eqid 2735 | . . . . 5 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
7 | eqid 2735 | . . . . 5 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
8 | lnosub.7 | . . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
9 | 2, 3, 4, 5, 6, 7, 8 | lnolin 30783 | . . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (-1 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) = ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝐵))( +𝑣 ‘𝑊)(𝑇‘𝐴))) |
10 | 1, 9 | mp3anr1 1457 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) = ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝐵))( +𝑣 ‘𝑊)(𝑇‘𝐴))) |
11 | 10 | ancom2s 650 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) = ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝐵))( +𝑣 ‘𝑊)(𝑇‘𝐴))) |
12 | lnosub.5 | . . . . . 6 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
13 | 2, 4, 6, 12 | nvmval2 30672 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = ((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) |
14 | 13 | 3expb 1119 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑀𝐵) = ((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) |
15 | 14 | 3ad2antl1 1184 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑀𝐵) = ((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) |
16 | 15 | fveq2d 6911 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑀𝐵)) = (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴))) |
17 | simpl2 1191 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑊 ∈ NrmCVec) | |
18 | 2, 3, 8 | lnof 30784 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊)) |
19 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
20 | ffvelcdm 7101 | . . . 4 ⊢ ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐴 ∈ 𝑋) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) | |
21 | 18, 19, 20 | syl2an 596 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) |
22 | simpr 484 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
23 | ffvelcdm 7101 | . . . 4 ⊢ ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐵 ∈ 𝑋) → (𝑇‘𝐵) ∈ (BaseSet‘𝑊)) | |
24 | 18, 22, 23 | syl2an 596 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘𝐵) ∈ (BaseSet‘𝑊)) |
25 | lnosub.6 | . . . 4 ⊢ 𝑁 = ( −𝑣 ‘𝑊) | |
26 | 3, 5, 7, 25 | nvmval2 30672 | . . 3 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝐴) ∈ (BaseSet‘𝑊) ∧ (𝑇‘𝐵) ∈ (BaseSet‘𝑊)) → ((𝑇‘𝐴)𝑁(𝑇‘𝐵)) = ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝐵))( +𝑣 ‘𝑊)(𝑇‘𝐴))) |
27 | 17, 21, 24, 26 | syl3anc 1370 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑇‘𝐴)𝑁(𝑇‘𝐵)) = ((-1( ·𝑠OLD ‘𝑊)(𝑇‘𝐵))( +𝑣 ‘𝑊)(𝑇‘𝐴))) |
28 | 11, 16, 27 | 3eqtr4d 2785 | 1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑀𝐵)) = ((𝑇‘𝐴)𝑁(𝑇‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 -cneg 11491 NrmCVeccnv 30613 +𝑣 cpv 30614 BaseSetcba 30615 ·𝑠OLD cns 30616 −𝑣 cnsb 30618 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-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-lno 30773 |
This theorem is referenced by: blometi 30832 blocnilem 30833 ubthlem2 30900 |
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