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Mirrors > Home > HSE Home > Th. List > spanun | Structured version Visualization version GIF version |
Description: The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanun | ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 3982 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (𝐴 ∪ 𝐵) = (if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) | |
2 | 1 | fveq2d 6450 | . . 3 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (span‘(𝐴 ∪ 𝐵)) = (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵))) |
3 | fveq2 6446 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (span‘𝐴) = (span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ))) | |
4 | 3 | oveq1d 6937 | . . 3 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → ((span‘𝐴) +ℋ (span‘𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵))) |
5 | 2, 4 | eqeq12d 2792 | . 2 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → ((span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵)) ↔ (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵)))) |
6 | uneq2 3983 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵) = (if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ))) | |
7 | 6 | fveq2d 6450 | . . 3 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) = (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ)))) |
8 | fveq2 6446 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (span‘𝐵) = (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ))) | |
9 | 8 | oveq2d 6938 | . . 3 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ)))) |
10 | 7, 9 | eqeq12d 2792 | . 2 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → ((span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵)) ↔ (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ))) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ))))) |
11 | sseq1 3844 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (𝐴 ⊆ ℋ ↔ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ)) | |
12 | sseq1 3844 | . . . 4 ⊢ ( ℋ = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → ( ℋ ⊆ ℋ ↔ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ)) | |
13 | ssid 3841 | . . . 4 ⊢ ℋ ⊆ ℋ | |
14 | 11, 12, 13 | elimhyp 4369 | . . 3 ⊢ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ |
15 | sseq1 3844 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (𝐵 ⊆ ℋ ↔ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ)) | |
16 | sseq1 3844 | . . . 4 ⊢ ( ℋ = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → ( ℋ ⊆ ℋ ↔ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ)) | |
17 | 15, 16, 13 | elimhyp 4369 | . . 3 ⊢ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ |
18 | 14, 17 | spanuni 28989 | . 2 ⊢ (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ))) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ))) |
19 | 5, 10, 18 | dedth2h 4363 | 1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∪ cun 3789 ⊆ wss 3791 ifcif 4306 ‘cfv 6135 (class class class)co 6922 ℋchba 28362 +ℋ cph 28374 spancspn 28375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 ax-hilex 28442 ax-hfvadd 28443 ax-hvcom 28444 ax-hvass 28445 ax-hv0cl 28446 ax-hvaddid 28447 ax-hfvmul 28448 ax-hvmulid 28449 ax-hvmulass 28450 ax-hvdistr1 28451 ax-hvdistr2 28452 ax-hvmul0 28453 ax-hfi 28522 ax-his1 28525 ax-his2 28526 ax-his3 28527 ax-his4 28528 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-icc 12494 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-top 21106 df-topon 21123 df-bases 21158 df-lm 21441 df-haus 21527 df-grpo 27934 df-gid 27935 df-ginv 27936 df-gdiv 27937 df-ablo 27986 df-vc 28000 df-nv 28033 df-va 28036 df-ba 28037 df-sm 28038 df-0v 28039 df-vs 28040 df-nmcv 28041 df-ims 28042 df-hnorm 28411 df-hvsub 28414 df-hlim 28415 df-sh 28650 df-ch 28664 df-ch0 28696 df-shs 28753 df-span 28754 |
This theorem is referenced by: spanpr 29025 superpos 29799 |
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