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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 31008 | . . . . 5 β’ Sβ β π« β | |
| 2 | sstr 3985 | . . . . 5 β’ ((π΄ β Sβ β§ Sβ β π« β) β π΄ β π« β) | |
| 3 | 1, 2 | mpan2 688 | . . . 4 β’ (π΄ β Sβ β π΄ β π« β) |
| 4 | 3 | unissd 4912 | . . 3 β’ (π΄ β Sβ β βͺ π΄ β βͺ π« β) |
| 5 | unipw 5443 | . . 3 β’ βͺ π« β = β | |
| 6 | 4, 5 | sseqtrdi 4027 | . 2 β’ (π΄ β Sβ β βͺ π΄ β β) |
| 7 | spanss2 31107 | . 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 3943 π« cpw 4597 βͺ cuni 4902 βcfv 6537 βchba 30681 Sβ csh 30690 spancspn 30694 |
| 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 7722 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172 ax-hilex 30761 ax-hfvadd 30762 ax-hv0cl 30765 ax-hfvmul 30767 |
| 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-ral 3056 df-rex 3065 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-map 8824 df-nn 12217 df-hlim 30734 df-sh 30969 df-ch 30983 df-span 31071 |
| This theorem is referenced by: (None) |
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