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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 12380 | . . . . 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 30653 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑄 ∈ 𝑋) |
| 6 | 5 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑄 ∈ 𝑋) |
| 7 | 2, 6, 6 | 3jca 1129 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (-1 ∈ ℂ ∧ 𝑄 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋)) |
| 8 | lno0.2 | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
| 9 | eqid 2737 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 10 | eqid 2737 | . . . 4 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
| 11 | eqid 2737 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
| 12 | eqid 2737 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
| 13 | lno0.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
| 14 | 3, 8, 9, 10, 11, 12, 13 | lnolin 30773 | . . 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 30671 | . . . . 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 6910 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = (𝑇‘𝑄)) |
| 19 | 18 | 3ad2ant1 1134 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘((-1( ·𝑠OLD ‘𝑈)𝑄)( +𝑣 ‘𝑈)𝑄)) = (𝑇‘𝑄)) |
| 20 | simp2 1138 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ NrmCVec) | |
| 21 | 3, 8, 13 | lnof 30774 | . . . 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 30671 | . . 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 2785 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 -cneg 11493 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 0veccn0v 30607 LnOp clno 30759 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 df-lno 30763 |
| This theorem is referenced by: lnomul 30779 nmlno0lem 30812 nmlnoubi 30815 lnon0 30817 nmblolbii 30818 blocnilem 30823 |
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