Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 0ellsp | Structured version Visualization version GIF version |
Description: Zero is in all spans. (Contributed by Thierry Arnoux, 8-May-2023.) |
Ref | Expression |
---|---|
0ellsp.1 | ⊢ 0 = (0g‘𝑊) |
0ellsp.b | ⊢ 𝐵 = (Base‘𝑊) |
0ellsp.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
0ellsp | ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ⊆ 𝐵) → 0 ∈ (𝑁‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ellsp.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | eqid 2820 | . . 3 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
3 | 0ellsp.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 1, 2, 3 | lspcl 19741 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ⊆ 𝐵) → (𝑁‘𝑆) ∈ (LSubSp‘𝑊)) |
5 | 0ellsp.1 | . . 3 ⊢ 0 = (0g‘𝑊) | |
6 | 5, 2 | lss0cl 19711 | . 2 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘𝑆) ∈ (LSubSp‘𝑊)) → 0 ∈ (𝑁‘𝑆)) |
7 | 4, 6 | syldan 593 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ⊆ 𝐵) → 0 ∈ (𝑁‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 ‘cfv 6348 Basecbs 16476 0gc0g 16706 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 |
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-1st 7682 df-2nd 7683 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 16479 df-slot 16480 df-base 16482 df-sets 16483 df-plusg 16571 df-0g 16708 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-grp 18099 df-minusg 18100 df-sbg 18101 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-lss 19697 df-lsp 19737 |
This theorem is referenced by: 0nellinds 30954 lbslsat 31036 lbsdiflsp0 31044 dimkerim 31045 |
Copyright terms: Public domain | W3C validator |