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Mirrors > Home > HSE Home > Th. List > spanpr | Structured version Visualization version GIF version |
Description: The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanpr | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (span‘{(𝐴 +ℎ 𝐵)}) ⊆ (span‘{𝐴, 𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spansnsh 30779 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (span‘{𝐴}) ∈ Sℋ ) | |
2 | spansnsh 30779 | . . . . . . 7 ⊢ (𝐵 ∈ ℋ → (span‘{𝐵}) ∈ Sℋ ) | |
3 | shscl 30536 | . . . . . . 7 ⊢ (((span‘{𝐴}) ∈ Sℋ ∧ (span‘{𝐵}) ∈ Sℋ ) → ((span‘{𝐴}) +ℋ (span‘{𝐵})) ∈ Sℋ ) | |
4 | 1, 2, 3 | syl2an 597 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((span‘{𝐴}) +ℋ (span‘{𝐵})) ∈ Sℋ ) |
5 | 4 | adantr 482 | . . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ (span‘{(𝐴 +ℎ 𝐵)})) → ((span‘{𝐴}) +ℋ (span‘{𝐵})) ∈ Sℋ ) |
6 | 1, 2 | anim12i 614 | . . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((span‘{𝐴}) ∈ Sℋ ∧ (span‘{𝐵}) ∈ Sℋ )) |
7 | spansnid 30781 | . . . . . . . 8 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (span‘{𝐴})) | |
8 | spansnid 30781 | . . . . . . . 8 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (span‘{𝐵})) | |
9 | 7, 8 | anim12i 614 | . . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ∈ (span‘{𝐴}) ∧ 𝐵 ∈ (span‘{𝐵}))) |
10 | shsva 30538 | . . . . . . 7 ⊢ (((span‘{𝐴}) ∈ Sℋ ∧ (span‘{𝐵}) ∈ Sℋ ) → ((𝐴 ∈ (span‘{𝐴}) ∧ 𝐵 ∈ (span‘{𝐵})) → (𝐴 +ℎ 𝐵) ∈ ((span‘{𝐴}) +ℋ (span‘{𝐵})))) | |
11 | 6, 9, 10 | sylc 65 | . . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ((span‘{𝐴}) +ℋ (span‘{𝐵}))) |
12 | 11 | adantr 482 | . . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ (span‘{(𝐴 +ℎ 𝐵)})) → (𝐴 +ℎ 𝐵) ∈ ((span‘{𝐴}) +ℋ (span‘{𝐵}))) |
13 | simpr 486 | . . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ (span‘{(𝐴 +ℎ 𝐵)})) → 𝑥 ∈ (span‘{(𝐴 +ℎ 𝐵)})) | |
14 | elspansn3 30790 | . . . . 5 ⊢ ((((span‘{𝐴}) +ℋ (span‘{𝐵})) ∈ Sℋ ∧ (𝐴 +ℎ 𝐵) ∈ ((span‘{𝐴}) +ℋ (span‘{𝐵})) ∧ 𝑥 ∈ (span‘{(𝐴 +ℎ 𝐵)})) → 𝑥 ∈ ((span‘{𝐴}) +ℋ (span‘{𝐵}))) | |
15 | 5, 12, 13, 14 | syl3anc 1372 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝑥 ∈ (span‘{(𝐴 +ℎ 𝐵)})) → 𝑥 ∈ ((span‘{𝐴}) +ℋ (span‘{𝐵}))) |
16 | 15 | ex 414 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑥 ∈ (span‘{(𝐴 +ℎ 𝐵)}) → 𝑥 ∈ ((span‘{𝐴}) +ℋ (span‘{𝐵})))) |
17 | 16 | ssrdv 3986 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (span‘{(𝐴 +ℎ 𝐵)}) ⊆ ((span‘{𝐴}) +ℋ (span‘{𝐵}))) |
18 | df-pr 4627 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
19 | 18 | fveq2i 6884 | . . 3 ⊢ (span‘{𝐴, 𝐵}) = (span‘({𝐴} ∪ {𝐵})) |
20 | snssi 4807 | . . . 4 ⊢ (𝐴 ∈ ℋ → {𝐴} ⊆ ℋ) | |
21 | snssi 4807 | . . . 4 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ) | |
22 | spanun 30763 | . . . 4 ⊢ (({𝐴} ⊆ ℋ ∧ {𝐵} ⊆ ℋ) → (span‘({𝐴} ∪ {𝐵})) = ((span‘{𝐴}) +ℋ (span‘{𝐵}))) | |
23 | 20, 21, 22 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (span‘({𝐴} ∪ {𝐵})) = ((span‘{𝐴}) +ℋ (span‘{𝐵}))) |
24 | 19, 23 | eqtr2id 2786 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((span‘{𝐴}) +ℋ (span‘{𝐵})) = (span‘{𝐴, 𝐵})) |
25 | 17, 24 | sseqtrd 4020 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (span‘{(𝐴 +ℎ 𝐵)}) ⊆ (span‘{𝐴, 𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3944 ⊆ wss 3946 {csn 4624 {cpr 4626 ‘cfv 6535 (class class class)co 7396 ℋchba 30137 +ℎ cva 30138 Sℋ csh 30146 +ℋ cph 30149 spancspn 30150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 ax-cc 10417 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 ax-addf 11176 ax-mulf 11177 ax-hilex 30217 ax-hfvadd 30218 ax-hvcom 30219 ax-hvass 30220 ax-hv0cl 30221 ax-hvaddid 30222 ax-hfvmul 30223 ax-hvmulid 30224 ax-hvmulass 30225 ax-hvdistr1 30226 ax-hvdistr2 30227 ax-hvmul0 30228 ax-hfi 30297 ax-his1 30300 ax-his2 30301 ax-his3 30302 ax-his4 30303 ax-hcompl 30420 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 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-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8691 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-fi 9393 df-sup 9424 df-inf 9425 df-oi 9492 df-card 9921 df-acn 9924 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13315 df-ico 13317 df-icc 13318 df-fz 13472 df-fzo 13615 df-fl 13744 df-seq 13954 df-exp 14015 df-hash 14278 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-clim 15419 df-rlim 15420 df-sum 15620 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-starv 17199 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-hom 17208 df-cco 17209 df-rest 17355 df-topn 17356 df-0g 17374 df-gsum 17375 df-topgen 17376 df-pt 17377 df-prds 17380 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-mulg 18936 df-cntz 19166 df-cmn 19634 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-fbas 20915 df-fg 20916 df-cnfld 20919 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-cld 22492 df-ntr 22493 df-cls 22494 df-nei 22571 df-cn 22700 df-cnp 22701 df-lm 22702 df-haus 22788 df-tx 23035 df-hmeo 23228 df-fil 23319 df-fm 23411 df-flim 23412 df-flf 23413 df-xms 23795 df-ms 23796 df-tms 23797 df-cfil 24741 df-cau 24742 df-cmet 24743 df-grpo 29711 df-gid 29712 df-ginv 29713 df-gdiv 29714 df-ablo 29763 df-vc 29777 df-nv 29810 df-va 29813 df-ba 29814 df-sm 29815 df-0v 29816 df-vs 29817 df-nmcv 29818 df-ims 29819 df-dip 29919 df-ssp 29940 df-ph 30031 df-cbn 30081 df-hnorm 30186 df-hba 30187 df-hvsub 30189 df-hlim 30190 df-hcau 30191 df-sh 30425 df-ch 30439 df-oc 30470 df-ch0 30471 df-shs 30526 df-span 30527 |
This theorem is referenced by: superpos 31572 |
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