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Mirrors > Home > HSE Home > Th. List > shscl | Structured version Visualization version GIF version |
Description: Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shscl | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ∈ Sℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7275 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Sℋ , 𝐴, ℋ) → (𝐴 +ℋ 𝐵) = (if(𝐴 ∈ Sℋ , 𝐴, ℋ) +ℋ 𝐵)) | |
2 | 1 | eleq1d 2824 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Sℋ , 𝐴, ℋ) → ((𝐴 +ℋ 𝐵) ∈ Sℋ ↔ (if(𝐴 ∈ Sℋ , 𝐴, ℋ) +ℋ 𝐵) ∈ Sℋ )) |
3 | oveq2 7276 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ Sℋ , 𝐵, ℋ) → (if(𝐴 ∈ Sℋ , 𝐴, ℋ) +ℋ 𝐵) = (if(𝐴 ∈ Sℋ , 𝐴, ℋ) +ℋ if(𝐵 ∈ Sℋ , 𝐵, ℋ))) | |
4 | 3 | eleq1d 2824 | . 2 ⊢ (𝐵 = if(𝐵 ∈ Sℋ , 𝐵, ℋ) → ((if(𝐴 ∈ Sℋ , 𝐴, ℋ) +ℋ 𝐵) ∈ Sℋ ↔ (if(𝐴 ∈ Sℋ , 𝐴, ℋ) +ℋ if(𝐵 ∈ Sℋ , 𝐵, ℋ)) ∈ Sℋ )) |
5 | helsh 29586 | . . . 4 ⊢ ℋ ∈ Sℋ | |
6 | 5 | elimel 4533 | . . 3 ⊢ if(𝐴 ∈ Sℋ , 𝐴, ℋ) ∈ Sℋ |
7 | 5 | elimel 4533 | . . 3 ⊢ if(𝐵 ∈ Sℋ , 𝐵, ℋ) ∈ Sℋ |
8 | 6, 7 | shscli 29658 | . 2 ⊢ (if(𝐴 ∈ Sℋ , 𝐴, ℋ) +ℋ if(𝐵 ∈ Sℋ , 𝐵, ℋ)) ∈ Sℋ |
9 | 2, 4, 8 | dedth2h 4523 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ∈ Sℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ifcif 4464 (class class class)co 7268 ℋchba 29260 Sℋ csh 29269 +ℋ cph 29272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-hilex 29340 ax-hfvadd 29341 ax-hvcom 29342 ax-hvass 29343 ax-hv0cl 29344 ax-hvaddid 29345 ax-hfvmul 29346 ax-hvmulid 29347 ax-hvdistr1 29349 ax-hvdistr2 29350 ax-hvmul0 29351 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-ltxr 10998 df-sub 11190 df-neg 11191 df-nn 11957 df-grpo 28834 df-ablo 28886 df-hvsub 29312 df-hlim 29313 df-sh 29548 df-ch 29562 df-shs 29649 |
This theorem is referenced by: shsvs 29664 spanpr 29921 chscllem2 29979 chscl 29982 |
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