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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 | β’ ((π΄ β β β§ π΅ β β) β (spanβ(π΄ βͺ π΅)) = ((spanβπ΄) +β (spanβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 4155 | . . . 4 β’ (π΄ = if(π΄ β β, π΄, β) β (π΄ βͺ π΅) = (if(π΄ β β, π΄, β) βͺ π΅)) | |
2 | 1 | fveq2d 6901 | . . 3 β’ (π΄ = if(π΄ β β, π΄, β) β (spanβ(π΄ βͺ π΅)) = (spanβ(if(π΄ β β, π΄, β) βͺ π΅))) |
3 | fveq2 6897 | . . . 4 β’ (π΄ = if(π΄ β β, π΄, β) β (spanβπ΄) = (spanβif(π΄ β β, π΄, β))) | |
4 | 3 | oveq1d 7435 | . . 3 β’ (π΄ = if(π΄ β β, π΄, β) β ((spanβπ΄) +β (spanβπ΅)) = ((spanβif(π΄ β β, π΄, β)) +β (spanβπ΅))) |
5 | 2, 4 | eqeq12d 2744 | . 2 β’ (π΄ = if(π΄ β β, π΄, β) β ((spanβ(π΄ βͺ π΅)) = ((spanβπ΄) +β (spanβπ΅)) β (spanβ(if(π΄ β β, π΄, β) βͺ π΅)) = ((spanβif(π΄ β β, π΄, β)) +β (spanβπ΅)))) |
6 | uneq2 4156 | . . . 4 β’ (π΅ = if(π΅ β β, π΅, β) β (if(π΄ β β, π΄, β) βͺ π΅) = (if(π΄ β β, π΄, β) βͺ if(π΅ β β, π΅, β))) | |
7 | 6 | fveq2d 6901 | . . 3 β’ (π΅ = if(π΅ β β, π΅, β) β (spanβ(if(π΄ β β, π΄, β) βͺ π΅)) = (spanβ(if(π΄ β β, π΄, β) βͺ if(π΅ β β, π΅, β)))) |
8 | fveq2 6897 | . . . 4 β’ (π΅ = if(π΅ β β, π΅, β) β (spanβπ΅) = (spanβif(π΅ β β, π΅, β))) | |
9 | 8 | oveq2d 7436 | . . 3 β’ (π΅ = if(π΅ β β, π΅, β) β ((spanβif(π΄ β β, π΄, β)) +β (spanβπ΅)) = ((spanβif(π΄ β β, π΄, β)) +β (spanβif(π΅ β β, π΅, β)))) |
10 | 7, 9 | eqeq12d 2744 | . 2 β’ (π΅ = if(π΅ β β, π΅, β) β ((spanβ(if(π΄ β β, π΄, β) βͺ π΅)) = ((spanβif(π΄ β β, π΄, β)) +β (spanβπ΅)) β (spanβ(if(π΄ β β, π΄, β) βͺ if(π΅ β β, π΅, β))) = ((spanβif(π΄ β β, π΄, β)) +β (spanβif(π΅ β β, π΅, β))))) |
11 | sseq1 4005 | . . . 4 β’ (π΄ = if(π΄ β β, π΄, β) β (π΄ β β β if(π΄ β β, π΄, β) β β)) | |
12 | sseq1 4005 | . . . 4 β’ ( β = if(π΄ β β, π΄, β) β ( β β β β if(π΄ β β, π΄, β) β β)) | |
13 | ssid 4002 | . . . 4 β’ β β β | |
14 | 11, 12, 13 | elimhyp 4594 | . . 3 β’ if(π΄ β β, π΄, β) β β |
15 | sseq1 4005 | . . . 4 β’ (π΅ = if(π΅ β β, π΅, β) β (π΅ β β β if(π΅ β β, π΅, β) β β)) | |
16 | sseq1 4005 | . . . 4 β’ ( β = if(π΅ β β, π΅, β) β ( β β β β if(π΅ β β, π΅, β) β β)) | |
17 | 15, 16, 13 | elimhyp 4594 | . . 3 β’ if(π΅ β β, π΅, β) β β |
18 | 14, 17 | spanuni 31367 | . 2 β’ (spanβ(if(π΄ β β, π΄, β) βͺ if(π΅ β β, π΅, β))) = ((spanβif(π΄ β β, π΄, β)) +β (spanβif(π΅ β β, π΅, β))) |
19 | 5, 10, 18 | dedth2h 4588 | 1 β’ ((π΄ β β β§ π΅ β β) β (spanβ(π΄ βͺ π΅)) = ((spanβπ΄) +β (spanβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 βͺ cun 3945 β wss 3947 ifcif 4529 βcfv 6548 (class class class)co 7420 βchba 30742 +β cph 30754 spancspn 30755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 ax-hilex 30822 ax-hfvadd 30823 ax-hvcom 30824 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvmulass 30830 ax-hvdistr1 30831 ax-hvdistr2 30832 ax-hvmul0 30833 ax-hfi 30902 ax-his1 30905 ax-his2 30906 ax-his3 30907 ax-his4 30908 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-icc 13364 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-topgen 17425 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-bases 22862 df-lm 23146 df-haus 23232 df-grpo 30316 df-gid 30317 df-ginv 30318 df-gdiv 30319 df-ablo 30368 df-vc 30382 df-nv 30415 df-va 30418 df-ba 30419 df-sm 30420 df-0v 30421 df-vs 30422 df-nmcv 30423 df-ims 30424 df-hnorm 30791 df-hvsub 30794 df-hlim 30795 df-sh 31030 df-ch 31044 df-ch0 31076 df-shs 31131 df-span 31132 |
This theorem is referenced by: spanpr 31403 superpos 32177 |
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