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Mirrors > Home > HSE Home > Th. List > shs0i | Structured version Visualization version GIF version |
Description: Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shne0.1 | ⊢ 𝐴 ∈ Sℋ |
Ref | Expression |
---|---|
shs0i | ⊢ (𝐴 +ℋ 0ℋ) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shne0.1 | . . 3 ⊢ 𝐴 ∈ Sℋ | |
2 | h0elsh 30372 | . . 3 ⊢ 0ℋ ∈ Sℋ | |
3 | 1, 2 | shsval3i 30504 | . 2 ⊢ (𝐴 +ℋ 0ℋ) = (span‘(𝐴 ∪ 0ℋ)) |
4 | sh0le 30556 | . . . . 5 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | |
5 | 1, 4 | ax-mp 5 | . . . 4 ⊢ 0ℋ ⊆ 𝐴 |
6 | ssequn2 4179 | . . . 4 ⊢ (0ℋ ⊆ 𝐴 ↔ (𝐴 ∪ 0ℋ) = 𝐴) | |
7 | 5, 6 | mpbi 229 | . . 3 ⊢ (𝐴 ∪ 0ℋ) = 𝐴 |
8 | 7 | fveq2i 6881 | . 2 ⊢ (span‘(𝐴 ∪ 0ℋ)) = (span‘𝐴) |
9 | spanid 30463 | . . 3 ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴) | |
10 | 1, 9 | ax-mp 5 | . 2 ⊢ (span‘𝐴) = 𝐴 |
11 | 3, 8, 10 | 3eqtri 2763 | 1 ⊢ (𝐴 +ℋ 0ℋ) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∪ cun 3942 ⊆ wss 3944 ‘cfv 6532 (class class class)co 7393 Sℋ csh 30044 +ℋ cph 30047 spancspn 30048 0ℋc0h 30051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 ax-hilex 30115 ax-hfvadd 30116 ax-hvcom 30117 ax-hvass 30118 ax-hv0cl 30119 ax-hvaddid 30120 ax-hfvmul 30121 ax-hvmulid 30122 ax-hvmulass 30123 ax-hvdistr1 30124 ax-hvdistr2 30125 ax-hvmul0 30126 ax-hfi 30195 ax-his1 30198 ax-his2 30199 ax-his3 30200 ax-his4 30201 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 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 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-map 8805 df-pm 8806 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-n0 12455 df-z 12541 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-icc 13313 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-topgen 17371 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-top 22325 df-topon 22342 df-bases 22378 df-lm 22662 df-haus 22748 df-grpo 29609 df-gid 29610 df-ginv 29611 df-gdiv 29612 df-ablo 29661 df-vc 29675 df-nv 29708 df-va 29711 df-ba 29712 df-sm 29713 df-0v 29714 df-vs 29715 df-nmcv 29716 df-ims 29717 df-hnorm 30084 df-hvsub 30087 df-hlim 30088 df-sh 30323 df-ch 30337 df-ch0 30369 df-shs 30424 df-span 30425 |
This theorem is referenced by: shs00i 30566 sumdmdlem2 31535 |
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