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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 11754 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | lnosub.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | eqid 2823 | . . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
4 | eqid 2823 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
5 | eqid 2823 | . . . . 5 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
6 | eqid 2823 | . . . . 5 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
7 | eqid 2823 | . . . . 5 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
8 | lnosub.7 | . . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
9 | 2, 3, 4, 5, 6, 7, 8 | lnolin 28533 | . . . 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 1454 | . . 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 648 | . 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 28422 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = ((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) |
14 | 13 | 3expb 1116 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑀𝐵) = ((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) |
15 | 14 | 3ad2antl1 1181 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑀𝐵) = ((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴)) |
16 | 15 | fveq2d 6676 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑀𝐵)) = (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝐵)( +𝑣 ‘𝑈)𝐴))) |
17 | simpl2 1188 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑊 ∈ NrmCVec) | |
18 | 2, 3, 8 | lnof 28534 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊)) |
19 | simpl 485 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
20 | ffvelrn 6851 | . . . 4 ⊢ ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐴 ∈ 𝑋) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) | |
21 | 18, 19, 20 | syl2an 597 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) |
22 | simpr 487 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
23 | ffvelrn 6851 | . . . 4 ⊢ ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐵 ∈ 𝑋) → (𝑇‘𝐵) ∈ (BaseSet‘𝑊)) | |
24 | 18, 22, 23 | syl2an 597 | . . 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 28422 | . . 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 1367 | . 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 2868 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 -cneg 10873 NrmCVeccnv 28363 +𝑣 cpv 28364 BaseSetcba 28365 ·𝑠OLD cns 28366 −𝑣 cnsb 28368 LnOp clno 28519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-lno 28523 |
This theorem is referenced by: blometi 28582 blocnilem 28583 ubthlem2 28650 |
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