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Mirrors > Home > HSE Home > Th. List > shsidmi | Structured version Visualization version GIF version |
Description: Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shsidm.1 | ⊢ 𝐴 ∈ Sℋ |
Ref | Expression |
---|---|
shsidmi | ⊢ (𝐴 +ℋ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shsidm.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
2 | 1, 1 | shseli 28730 | . . . 4 ⊢ (𝑥 ∈ (𝐴 +ℋ 𝐴) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 +ℎ 𝑧)) |
3 | shaddcl 28629 | . . . . . . 7 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 +ℎ 𝑧) ∈ 𝐴) | |
4 | 1, 3 | mp3an1 1578 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 +ℎ 𝑧) ∈ 𝐴) |
5 | eleq1 2894 | . . . . . 6 ⊢ (𝑥 = (𝑦 +ℎ 𝑧) → (𝑥 ∈ 𝐴 ↔ (𝑦 +ℎ 𝑧) ∈ 𝐴)) | |
6 | 4, 5 | syl5ibrcom 239 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴)) |
7 | 6 | rexlimivv 3246 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴) |
8 | 2, 7 | sylbi 209 | . . 3 ⊢ (𝑥 ∈ (𝐴 +ℋ 𝐴) → 𝑥 ∈ 𝐴) |
9 | 8 | ssriv 3831 | . 2 ⊢ (𝐴 +ℋ 𝐴) ⊆ 𝐴 |
10 | 1, 1 | shsub1i 28786 | . 2 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐴) |
11 | 9, 10 | eqssi 3843 | 1 ⊢ (𝐴 +ℋ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3118 (class class class)co 6905 +ℎ cva 28332 Sℋ csh 28340 +ℋ cph 28343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-hilex 28411 ax-hfvadd 28412 ax-hvcom 28413 ax-hvass 28414 ax-hv0cl 28415 ax-hvaddid 28416 ax-hfvmul 28417 ax-hvmulid 28418 ax-hvdistr2 28421 ax-hvmul0 28422 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-ltxr 10396 df-sub 10587 df-neg 10588 df-grpo 27903 df-ablo 27955 df-hvsub 28383 df-sh 28619 df-shs 28722 |
This theorem is referenced by: shslubi 28799 |
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