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Mirrors > Home > MPE Home > Th. List > lsslsp | Structured version Visualization version GIF version |
Description: Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap 𝑀‘𝐺 and 𝑁‘𝐺 since we are computing a property of 𝑁‘𝐺? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015. |
Ref | Expression |
---|---|
lsslsp.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
lsslsp.m | ⊢ 𝑀 = (LSpan‘𝑊) |
lsslsp.n | ⊢ 𝑁 = (LSpan‘𝑋) |
lsslsp.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lsslsp | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) = (𝑁‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝑊 ∈ LMod) | |
2 | lsslsp.x | . . . . . . . 8 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
3 | lsslsp.l | . . . . . . . 8 ⊢ 𝐿 = (LSubSp‘𝑊) | |
4 | 2, 3 | lsslmod 19732 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑋 ∈ LMod) |
5 | 4 | 3adant3 1128 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝑋 ∈ LMod) |
6 | simp3 1134 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝐺 ⊆ 𝑈) | |
7 | eqid 2821 | . . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
8 | 7, 3 | lssss 19708 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ (Base‘𝑊)) |
9 | 8 | 3ad2ant2 1130 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝑈 ⊆ (Base‘𝑊)) |
10 | 2, 7 | ressbas2 16555 | . . . . . . . 8 ⊢ (𝑈 ⊆ (Base‘𝑊) → 𝑈 = (Base‘𝑋)) |
11 | 9, 10 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝑈 = (Base‘𝑋)) |
12 | 6, 11 | sseqtrd 4007 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝐺 ⊆ (Base‘𝑋)) |
13 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
14 | eqid 2821 | . . . . . . 7 ⊢ (LSubSp‘𝑋) = (LSubSp‘𝑋) | |
15 | lsslsp.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑋) | |
16 | 13, 14, 15 | lspcl 19748 | . . . . . 6 ⊢ ((𝑋 ∈ LMod ∧ 𝐺 ⊆ (Base‘𝑋)) → (𝑁‘𝐺) ∈ (LSubSp‘𝑋)) |
17 | 5, 12, 16 | syl2anc 586 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑁‘𝐺) ∈ (LSubSp‘𝑋)) |
18 | 2, 3, 14 | lsslss 19733 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((𝑁‘𝐺) ∈ (LSubSp‘𝑋) ↔ ((𝑁‘𝐺) ∈ 𝐿 ∧ (𝑁‘𝐺) ⊆ 𝑈))) |
19 | 18 | 3adant3 1128 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → ((𝑁‘𝐺) ∈ (LSubSp‘𝑋) ↔ ((𝑁‘𝐺) ∈ 𝐿 ∧ (𝑁‘𝐺) ⊆ 𝑈))) |
20 | 17, 19 | mpbid 234 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → ((𝑁‘𝐺) ∈ 𝐿 ∧ (𝑁‘𝐺) ⊆ 𝑈)) |
21 | 20 | simpld 497 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑁‘𝐺) ∈ 𝐿) |
22 | 13, 15 | lspssid 19757 | . . . 4 ⊢ ((𝑋 ∈ LMod ∧ 𝐺 ⊆ (Base‘𝑋)) → 𝐺 ⊆ (𝑁‘𝐺)) |
23 | 5, 12, 22 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝐺 ⊆ (𝑁‘𝐺)) |
24 | lsslsp.m | . . . 4 ⊢ 𝑀 = (LSpan‘𝑊) | |
25 | 3, 24 | lspssp 19760 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘𝐺) ∈ 𝐿 ∧ 𝐺 ⊆ (𝑁‘𝐺)) → (𝑀‘𝐺) ⊆ (𝑁‘𝐺)) |
26 | 1, 21, 23, 25 | syl3anc 1367 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) ⊆ (𝑁‘𝐺)) |
27 | 6, 9 | sstrd 3977 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝐺 ⊆ (Base‘𝑊)) |
28 | 7, 3, 24 | lspcl 19748 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ⊆ (Base‘𝑊)) → (𝑀‘𝐺) ∈ 𝐿) |
29 | 1, 27, 28 | syl2anc 586 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) ∈ 𝐿) |
30 | 3, 24 | lspssp 19760 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) ⊆ 𝑈) |
31 | 2, 3, 14 | lsslss 19733 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → ((𝑀‘𝐺) ∈ (LSubSp‘𝑋) ↔ ((𝑀‘𝐺) ∈ 𝐿 ∧ (𝑀‘𝐺) ⊆ 𝑈))) |
32 | 31 | 3adant3 1128 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → ((𝑀‘𝐺) ∈ (LSubSp‘𝑋) ↔ ((𝑀‘𝐺) ∈ 𝐿 ∧ (𝑀‘𝐺) ⊆ 𝑈))) |
33 | 29, 30, 32 | mpbir2and 711 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) ∈ (LSubSp‘𝑋)) |
34 | 7, 24 | lspssid 19757 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ⊆ (Base‘𝑊)) → 𝐺 ⊆ (𝑀‘𝐺)) |
35 | 1, 27, 34 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → 𝐺 ⊆ (𝑀‘𝐺)) |
36 | 14, 15 | lspssp 19760 | . . 3 ⊢ ((𝑋 ∈ LMod ∧ (𝑀‘𝐺) ∈ (LSubSp‘𝑋) ∧ 𝐺 ⊆ (𝑀‘𝐺)) → (𝑁‘𝐺) ⊆ (𝑀‘𝐺)) |
37 | 5, 33, 35, 36 | syl3anc 1367 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑁‘𝐺) ⊆ (𝑀‘𝐺)) |
38 | 26, 37 | eqssd 3984 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) = (𝑁‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-sca 16581 df-vsca 16582 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 df-lsp 19744 |
This theorem is referenced by: lss0v 19788 lsslindf 20974 islinds3 20978 lbslsat 31014 dimkerim 31023 lcdlsp 38772 islssfg 39690 |
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