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Mirrors > Home > HSE Home > Th. List > shs00i | Structured version Visualization version GIF version |
Description: Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shne0.1 | ⊢ 𝐴 ∈ Sℋ |
shs00.2 | ⊢ 𝐵 ∈ Sℋ |
Ref | Expression |
---|---|
shs00i | ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) ↔ (𝐴 +ℋ 𝐵) = 0ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7362 | . . 3 ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) → (𝐴 +ℋ 𝐵) = (0ℋ +ℋ 0ℋ)) | |
2 | h0elsh 30084 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
3 | 2 | shs0i 30277 | . . 3 ⊢ (0ℋ +ℋ 0ℋ) = 0ℋ |
4 | 1, 3 | eqtrdi 2792 | . 2 ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) → (𝐴 +ℋ 𝐵) = 0ℋ) |
5 | shne0.1 | . . . . . 6 ⊢ 𝐴 ∈ Sℋ | |
6 | shs00.2 | . . . . . 6 ⊢ 𝐵 ∈ Sℋ | |
7 | 5, 6 | shsub1i 30200 | . . . . 5 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵) |
8 | sseq2 3968 | . . . . 5 ⊢ ((𝐴 +ℋ 𝐵) = 0ℋ → (𝐴 ⊆ (𝐴 +ℋ 𝐵) ↔ 𝐴 ⊆ 0ℋ)) | |
9 | 7, 8 | mpbii 232 | . . . 4 ⊢ ((𝐴 +ℋ 𝐵) = 0ℋ → 𝐴 ⊆ 0ℋ) |
10 | shle0 30270 | . . . . 5 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) | |
11 | 5, 10 | ax-mp 5 | . . . 4 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ) |
12 | 9, 11 | sylib 217 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) = 0ℋ → 𝐴 = 0ℋ) |
13 | 6, 5 | shsub2i 30201 | . . . . 5 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
14 | sseq2 3968 | . . . . 5 ⊢ ((𝐴 +ℋ 𝐵) = 0ℋ → (𝐵 ⊆ (𝐴 +ℋ 𝐵) ↔ 𝐵 ⊆ 0ℋ)) | |
15 | 13, 14 | mpbii 232 | . . . 4 ⊢ ((𝐴 +ℋ 𝐵) = 0ℋ → 𝐵 ⊆ 0ℋ) |
16 | shle0 30270 | . . . . 5 ⊢ (𝐵 ∈ Sℋ → (𝐵 ⊆ 0ℋ ↔ 𝐵 = 0ℋ)) | |
17 | 6, 16 | ax-mp 5 | . . . 4 ⊢ (𝐵 ⊆ 0ℋ ↔ 𝐵 = 0ℋ) |
18 | 15, 17 | sylib 217 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) = 0ℋ → 𝐵 = 0ℋ) |
19 | 12, 18 | jca 512 | . 2 ⊢ ((𝐴 +ℋ 𝐵) = 0ℋ → (𝐴 = 0ℋ ∧ 𝐵 = 0ℋ)) |
20 | 4, 19 | impbii 208 | 1 ⊢ ((𝐴 = 0ℋ ∧ 𝐵 = 0ℋ) ↔ (𝐴 +ℋ 𝐵) = 0ℋ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 (class class class)co 7353 Sℋ csh 29756 +ℋ cph 29759 0ℋc0h 29763 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 ax-addf 11126 ax-mulf 11127 ax-hilex 29827 ax-hfvadd 29828 ax-hvcom 29829 ax-hvass 29830 ax-hv0cl 29831 ax-hvaddid 29832 ax-hfvmul 29833 ax-hvmulid 29834 ax-hvmulass 29835 ax-hvdistr1 29836 ax-hvdistr2 29837 ax-hvmul0 29838 ax-hfi 29907 ax-his1 29910 ax-his2 29911 ax-his3 29912 ax-his4 29913 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-map 8763 df-pm 8764 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-inf 9375 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-n0 12410 df-z 12496 df-uz 12760 df-q 12866 df-rp 12908 df-xneg 13025 df-xadd 13026 df-xmul 13027 df-icc 13263 df-seq 13899 df-exp 13960 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-topgen 17317 df-psmet 20773 df-xmet 20774 df-met 20775 df-bl 20776 df-mopn 20777 df-top 22227 df-topon 22244 df-bases 22280 df-lm 22564 df-haus 22650 df-grpo 29321 df-gid 29322 df-ginv 29323 df-gdiv 29324 df-ablo 29373 df-vc 29387 df-nv 29420 df-va 29423 df-ba 29424 df-sm 29425 df-0v 29426 df-vs 29427 df-nmcv 29428 df-ims 29429 df-hnorm 29796 df-hvsub 29799 df-hlim 29800 df-sh 30035 df-ch 30049 df-ch0 30081 df-shs 30136 df-span 30137 |
This theorem is referenced by: (None) |
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