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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2n | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38671. (Contributed by NM, 12-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2m.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2m.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lclkrlem2m.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lclkrlem2m.q | ⊢ × = (.r‘𝑆) |
lclkrlem2m.z | ⊢ 0 = (0g‘𝑆) |
lclkrlem2m.i | ⊢ 𝐼 = (invr‘𝑆) |
lclkrlem2m.m | ⊢ − = (-g‘𝑈) |
lclkrlem2m.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2m.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2m.p | ⊢ + = (+g‘𝐷) |
lclkrlem2m.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lclkrlem2m.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lclkrlem2m.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lclkrlem2m.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lclkrlem2n.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2n.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2n.w | ⊢ (𝜑 → 𝑈 ∈ LVec) |
lclkrlem2n.j | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 ) |
lclkrlem2n.k | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 ) |
Ref | Expression |
---|---|
lclkrlem2n | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
2 | lclkrlem2n.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | lclkrlem2n.w | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) | |
4 | lveclmod 19880 | . . 3 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑈 ∈ LMod) |
6 | lclkrlem2m.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lclkrlem2m.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lclkrlem2m.p | . . . 4 ⊢ + = (+g‘𝐷) | |
9 | lclkrlem2m.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
10 | lclkrlem2m.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 6, 7, 8, 5, 9, 10 | ldualvaddcl 36268 | . . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹) |
12 | lclkrlem2n.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
13 | 6, 12, 1 | lkrlss 36233 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈)) |
14 | 5, 11, 13 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ (LSubSp‘𝑈)) |
15 | lclkrlem2n.j | . . 3 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 ) | |
16 | lclkrlem2m.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
17 | lclkrlem2m.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑈) | |
18 | lclkrlem2m.z | . . . 4 ⊢ 0 = (0g‘𝑆) | |
19 | lclkrlem2m.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
20 | 16, 17, 18, 6, 12, 3, 11, 19 | ellkr2 36229 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘(𝐸 + 𝐺)) ↔ ((𝐸 + 𝐺)‘𝑋) = 0 )) |
21 | 15, 20 | mpbird 259 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐿‘(𝐸 + 𝐺))) |
22 | lclkrlem2n.k | . . 3 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 ) | |
23 | lclkrlem2m.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
24 | 16, 17, 18, 6, 12, 3, 11, 23 | ellkr2 36229 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝐿‘(𝐸 + 𝐺)) ↔ ((𝐸 + 𝐺)‘𝑌) = 0 )) |
25 | 22, 24 | mpbird 259 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐿‘(𝐸 + 𝐺))) |
26 | 1, 2, 5, 14, 21, 25 | lspprss 19766 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {cpr 4571 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 -gcsg 18107 invrcinvr 19423 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LVecclvec 19876 LFnlclfn 36195 LKerclk 36223 LDualcld 36261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-sca 16583 df-vsca 16584 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lfl 36196 df-lkr 36224 df-ldual 36262 |
This theorem is referenced by: lclkrlem2v 38666 |
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