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| Mirrors > Home > HSE Home > Th. List > shsupunss | Structured version Visualization version GIF version | ||
| Description: The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| shsupunss | β’ (π΄ β Sβ β βͺ π΄ β (spanββͺ π΄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | shsspwh 31072 | . . . . 5 β’ Sβ β π« β | |
| 2 | sstr 3980 | . . . . 5 β’ ((π΄ β Sβ β§ Sβ β π« β) β π΄ β π« β) | |
| 3 | 1, 2 | mpan2 689 | . . . 4 β’ (π΄ β Sβ β π΄ β π« β) |
| 4 | 3 | unissd 4911 | . . 3 β’ (π΄ β Sβ β βͺ π΄ β βͺ π« β) |
| 5 | unipw 5444 | . . 3 β’ βͺ π« β = β | |
| 6 | 4, 5 | sseqtrdi 4022 | . 2 β’ (π΄ β Sβ β βͺ π΄ β β) |
| 7 | spanss2 31171 | . 2 β’ (βͺ π΄ β β β βͺ π΄ β (spanββͺ π΄)) | |
| 8 | 6, 7 | syl 17 | 1 β’ (π΄ β Sβ β βͺ π΄ β (spanββͺ π΄)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wss 3939 π« cpw 4596 βͺ cuni 4901 βcfv 6541 βchba 30745 Sβ csh 30754 spancspn 30758 |
| 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 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-1cn 11194 ax-addcl 11196 ax-hilex 30825 ax-hfvadd 30826 ax-hv0cl 30829 ax-hfvmul 30831 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-map 8843 df-nn 12241 df-hlim 30798 df-sh 31033 df-ch 31047 df-span 31135 |
| This theorem is referenced by: (None) |
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