Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatn0 | Structured version Visualization version GIF version |
Description: A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 30120 analog.) (Contributed by NM, 25-Aug-2014.) |
Ref | Expression |
---|---|
lsatn0.o | ⊢ 0 = (0g‘𝑊) |
lsatn0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatn0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsatn0.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatn0 | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatn0.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
2 | lsatn0.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | eqid 2820 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2820 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | lsatn0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
6 | lsatn0.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | 3, 4, 5, 6 | islsat 36160 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
9 | 1, 8 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifsn 4712 | . . . . 5 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
11 | 3, 5, 4 | lspsneq0 19779 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
12 | 2, 11 | sylan 582 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
13 | 12 | biimpd 231 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } → 𝑣 = 0 )) |
14 | 13 | necon3d 3036 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑣 ≠ 0 → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
15 | 14 | expimpd 456 | . . . . 5 ⊢ (𝜑 → ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
16 | 10, 15 | syl5bi 244 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
17 | neeq1 3077 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 ≠ { 0 } ↔ ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) | |
18 | 17 | biimprcd 252 | . . . 4 ⊢ (((LSpan‘𝑊)‘{𝑣}) ≠ { 0 } → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 })) |
19 | 16, 18 | syl6 35 | . . 3 ⊢ (𝜑 → (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 }))) |
20 | 19 | rexlimdv 3282 | . 2 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 })) |
21 | 9, 20 | mpd 15 | 1 ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∃wrex 3138 ∖ cdif 3926 {csn 4560 ‘cfv 6348 Basecbs 16478 0gc0g 16708 LModclmod 19629 LSpanclspn 19738 LSAtomsclsa 36143 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-plusg 16573 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-mgp 19235 df-ring 19294 df-lmod 19631 df-lss 19699 df-lsp 19739 df-lsatoms 36145 |
This theorem is referenced by: lsatspn0 36169 lsatssn0 36171 lsatcmp 36172 lsatcv0 36200 dochsat 38552 dochsatshp 38620 dochshpsat 38623 dochexmidlem1 38629 |
Copyright terms: Public domain | W3C validator |