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Mirrors > Home > MPE Home > Th. List > lnon0 | Structured version Visualization version GIF version |
Description: The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnon0.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
lnon0.6 | ⊢ 𝑍 = (0vec‘𝑈) |
lnon0.0 | ⊢ 𝑂 = (𝑈 0op 𝑊) |
lnon0.7 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
Ref | Expression |
---|---|
lnon0 | ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralnex 3200 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ≠ 𝑍 ↔ ¬ ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍) | |
2 | nne 2988 | . . . . . 6 ⊢ (¬ 𝑥 ≠ 𝑍 ↔ 𝑥 = 𝑍) | |
3 | 2 | ralbii 3132 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ≠ 𝑍 ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍) |
4 | 1, 3 | bitr3i 278 | . . . 4 ⊢ (¬ ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍 ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍) |
5 | fveq2 6543 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑍 → (𝑇‘𝑥) = (𝑇‘𝑍)) | |
6 | lnon0.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
7 | eqid 2795 | . . . . . . . . . . 11 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
8 | lnon0.6 | . . . . . . . . . . 11 ⊢ 𝑍 = (0vec‘𝑈) | |
9 | eqid 2795 | . . . . . . . . . . 11 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
10 | lnon0.7 | . . . . . . . . . . 11 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
11 | 6, 7, 8, 9, 10 | lno0 28229 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑍) = (0vec‘𝑊)) |
12 | 5, 11 | sylan9eqr 2853 | . . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑥 = 𝑍) → (𝑇‘𝑥) = (0vec‘𝑊)) |
13 | 12 | ex 413 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑥 = 𝑍 → (𝑇‘𝑥) = (0vec‘𝑊))) |
14 | 13 | ralimdv 3145 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊))) |
15 | 6, 7, 10 | lnof 28228 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊)) |
16 | 15 | ffnd 6388 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇 Fn 𝑋) |
17 | 14, 16 | jctild 526 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → (𝑇 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊)))) |
18 | fconstfv 6846 | . . . . . . 7 ⊢ (𝑇:𝑋⟶{(0vec‘𝑊)} ↔ (𝑇 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊))) | |
19 | fvex 6556 | . . . . . . . 8 ⊢ (0vec‘𝑊) ∈ V | |
20 | 19 | fconst2 6839 | . . . . . . 7 ⊢ (𝑇:𝑋⟶{(0vec‘𝑊)} ↔ 𝑇 = (𝑋 × {(0vec‘𝑊)})) |
21 | 18, 20 | bitr3i 278 | . . . . . 6 ⊢ ((𝑇 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊)) ↔ 𝑇 = (𝑋 × {(0vec‘𝑊)})) |
22 | 17, 21 | syl6ib 252 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → 𝑇 = (𝑋 × {(0vec‘𝑊)}))) |
23 | lnon0.0 | . . . . . . . 8 ⊢ 𝑂 = (𝑈 0op 𝑊) | |
24 | 6, 9, 23 | 0ofval 28260 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {(0vec‘𝑊)})) |
25 | 24 | 3adant3 1125 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑂 = (𝑋 × {(0vec‘𝑊)})) |
26 | 25 | eqeq2d 2805 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 = 𝑂 ↔ 𝑇 = (𝑋 × {(0vec‘𝑊)}))) |
27 | 22, 26 | sylibrd 260 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → 𝑇 = 𝑂)) |
28 | 4, 27 | syl5bi 243 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍 → 𝑇 = 𝑂)) |
29 | 28 | necon1ad 3001 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑂 → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍)) |
30 | 29 | imp 407 | 1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∃wrex 3106 {csn 4476 × cxp 5446 Fn wfn 6225 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 NrmCVeccnv 28057 BaseSetcba 28059 0veccn0v 28061 LnOp clno 28213 0op c0o 28216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-po 5367 df-so 5368 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-1st 7550 df-2nd 7551 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-pnf 10528 df-mnf 10529 df-ltxr 10531 df-sub 10724 df-neg 10725 df-grpo 27966 df-gid 27967 df-ginv 27968 df-ablo 28018 df-vc 28032 df-nv 28065 df-va 28068 df-ba 28069 df-sm 28070 df-0v 28071 df-nmcv 28073 df-lno 28217 df-0o 28220 |
This theorem is referenced by: (None) |
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