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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 4151 | . . . 4 β’ (π΄ = if(π΄ β β, π΄, β) β (π΄ βͺ π΅) = (if(π΄ β β, π΄, β) βͺ π΅)) | |
2 | 1 | fveq2d 6888 | . . 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 6884 | . . . 4 β’ (π΄ = if(π΄ β β, π΄, β) β (spanβπ΄) = (spanβif(π΄ β β, π΄, β))) | |
4 | 3 | oveq1d 7419 | . . 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 2742 | . 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 4152 | . . . 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 6888 | . . 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 6884 | . . . 4 β’ (π΅ = if(π΅ β β, π΅, β) β (spanβπ΅) = (spanβif(π΅ β β, π΅, β))) | |
9 | 8 | oveq2d 7420 | . . 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 2742 | . 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 4002 | . . . 4 β’ (π΄ = if(π΄ β β, π΄, β) β (π΄ β β β if(π΄ β β, π΄, β) β β)) | |
12 | sseq1 4002 | . . . 4 β’ ( β = if(π΄ β β, π΄, β) β ( β β β β if(π΄ β β, π΄, β) β β)) | |
13 | ssid 3999 | . . . 4 β’ β β β | |
14 | 11, 12, 13 | elimhyp 4588 | . . 3 β’ if(π΄ β β, π΄, β) β β |
15 | sseq1 4002 | . . . 4 β’ (π΅ = if(π΅ β β, π΅, β) β (π΅ β β β if(π΅ β β, π΅, β) β β)) | |
16 | sseq1 4002 | . . . 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 4588 | . . 3 β’ if(π΅ β β, π΅, β) β β |
18 | 14, 17 | spanuni 31302 | . 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 4582 | 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 1533 βͺ cun 3941 β wss 3943 ifcif 4523 βcfv 6536 (class class class)co 7404 βchba 30677 +β cph 30689 spancspn 30690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 ax-hilex 30757 ax-hfvadd 30758 ax-hvcom 30759 ax-hvass 30760 ax-hv0cl 30761 ax-hvaddid 30762 ax-hfvmul 30763 ax-hvmulid 30764 ax-hvmulass 30765 ax-hvdistr1 30766 ax-hvdistr2 30767 ax-hvmul0 30768 ax-hfi 30837 ax-his1 30840 ax-his2 30841 ax-his3 30842 ax-his4 30843 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-icc 13334 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-topgen 17396 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-top 22747 df-topon 22764 df-bases 22800 df-lm 23084 df-haus 23170 df-grpo 30251 df-gid 30252 df-ginv 30253 df-gdiv 30254 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-sm 30355 df-0v 30356 df-vs 30357 df-nmcv 30358 df-ims 30359 df-hnorm 30726 df-hvsub 30729 df-hlim 30730 df-sh 30965 df-ch 30979 df-ch0 31011 df-shs 31066 df-span 31067 |
This theorem is referenced by: spanpr 31338 superpos 32112 |
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