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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 4132 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (𝐴 ∪ 𝐵) = (if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) | |
2 | 1 | fveq2d 6669 | . . 3 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (span‘(𝐴 ∪ 𝐵)) = (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵))) |
3 | fveq2 6665 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (span‘𝐴) = (span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ))) | |
4 | 3 | oveq1d 7165 | . . 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 2837 | . 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 4133 | . . . 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 6669 | . . 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 6665 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (span‘𝐵) = (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ))) | |
9 | 8 | oveq2d 7166 | . . 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 2837 | . 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 3992 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (𝐴 ⊆ ℋ ↔ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ)) | |
12 | sseq1 3992 | . . . 4 ⊢ ( ℋ = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → ( ℋ ⊆ ℋ ↔ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ)) | |
13 | ssid 3989 | . . . 4 ⊢ ℋ ⊆ ℋ | |
14 | 11, 12, 13 | elimhyp 4530 | . . 3 ⊢ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ |
15 | sseq1 3992 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (𝐵 ⊆ ℋ ↔ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ)) | |
16 | sseq1 3992 | . . . 4 ⊢ ( ℋ = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → ( ℋ ⊆ ℋ ↔ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ)) | |
17 | 15, 16, 13 | elimhyp 4530 | . . 3 ⊢ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ |
18 | 14, 17 | spanuni 29315 | . 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 4524 | 1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∪ cun 3934 ⊆ wss 3936 ifcif 4467 ‘cfv 6350 (class class class)co 7150 ℋchba 28690 +ℋ cph 28702 spancspn 28703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-lm 21831 df-haus 21917 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 df-ims 28372 df-hnorm 28739 df-hvsub 28742 df-hlim 28743 df-sh 28978 df-ch 28992 df-ch0 29024 df-shs 29079 df-span 29080 |
This theorem is referenced by: spanpr 29351 superpos 30125 |
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