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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 3067 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ≠ 𝑍 ↔ ¬ ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍) | |
2 | nne 2939 | . . . . . 6 ⊢ (¬ 𝑥 ≠ 𝑍 ↔ 𝑥 = 𝑍) | |
3 | 2 | ralbii 3088 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ≠ 𝑍 ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍) |
4 | 1, 3 | bitr3i 277 | . . . 4 ⊢ (¬ ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍 ↔ ∀𝑥 ∈ 𝑋 𝑥 = 𝑍) |
5 | fveq2 6891 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑍 → (𝑇‘𝑥) = (𝑇‘𝑍)) | |
6 | lnon0.1 | . . . . . . . . . . 11 ⊢ 𝑋 = (BaseSet‘𝑈) | |
7 | eqid 2727 | . . . . . . . . . . 11 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
8 | lnon0.6 | . . . . . . . . . . 11 ⊢ 𝑍 = (0vec‘𝑈) | |
9 | eqid 2727 | . . . . . . . . . . 11 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
10 | lnon0.7 | . . . . . . . . . . 11 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
11 | 6, 7, 8, 9, 10 | lno0 30540 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑍) = (0vec‘𝑊)) |
12 | 5, 11 | sylan9eqr 2789 | . . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑥 = 𝑍) → (𝑇‘𝑥) = (0vec‘𝑊)) |
13 | 12 | ex 412 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑥 = 𝑍 → (𝑇‘𝑥) = (0vec‘𝑊))) |
14 | 13 | ralimdv 3164 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊))) |
15 | 6, 7, 10 | lnof 30539 | . . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊)) |
16 | 15 | ffnd 6717 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇 Fn 𝑋) |
17 | 14, 16 | jctild 525 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → (𝑇 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊)))) |
18 | fconstfv 7218 | . . . . . . 7 ⊢ (𝑇:𝑋⟶{(0vec‘𝑊)} ↔ (𝑇 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊))) | |
19 | fvex 6904 | . . . . . . . 8 ⊢ (0vec‘𝑊) ∈ V | |
20 | 19 | fconst2 7211 | . . . . . . 7 ⊢ (𝑇:𝑋⟶{(0vec‘𝑊)} ↔ 𝑇 = (𝑋 × {(0vec‘𝑊)})) |
21 | 18, 20 | bitr3i 277 | . . . . . 6 ⊢ ((𝑇 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (0vec‘𝑊)) ↔ 𝑇 = (𝑋 × {(0vec‘𝑊)})) |
22 | 17, 21 | imbitrdi 250 | . . . . 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 30571 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {(0vec‘𝑊)})) |
25 | 24 | 3adant3 1130 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑂 = (𝑋 × {(0vec‘𝑊)})) |
26 | 25 | eqeq2d 2738 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 = 𝑂 ↔ 𝑇 = (𝑋 × {(0vec‘𝑊)}))) |
27 | 22, 26 | sylibrd 259 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (∀𝑥 ∈ 𝑋 𝑥 = 𝑍 → 𝑇 = 𝑂)) |
28 | 4, 27 | biimtrid 241 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍 → 𝑇 = 𝑂)) |
29 | 28 | necon1ad 2952 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑂 → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍)) |
30 | 29 | imp 406 | 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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 {csn 4624 × cxp 5670 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 NrmCVeccnv 30368 BaseSetcba 30370 0veccn0v 30372 LnOp clno 30524 0op c0o 30527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-ltxr 11269 df-sub 11462 df-neg 11463 df-grpo 30277 df-gid 30278 df-ginv 30279 df-ablo 30329 df-vc 30343 df-nv 30376 df-va 30379 df-ba 30380 df-sm 30381 df-0v 30382 df-nmcv 30384 df-lno 30528 df-0o 30531 |
This theorem is referenced by: (None) |
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