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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 31198 | . . . 4 ⊢ (𝑥 ∈ (𝐴 +ℋ 𝐴) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 +ℎ 𝑧)) |
3 | shaddcl 31099 | . . . . . . 7 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 +ℎ 𝑧) ∈ 𝐴) | |
4 | 1, 3 | mp3an1 1444 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦 +ℎ 𝑧) ∈ 𝐴) |
5 | eleq1 2813 | . . . . . 6 ⊢ (𝑥 = (𝑦 +ℎ 𝑧) → (𝑥 ∈ 𝐴 ↔ (𝑦 +ℎ 𝑧) ∈ 𝐴)) | |
6 | 4, 5 | syl5ibrcom 246 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴)) |
7 | 6 | rexlimivv 3189 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴) |
8 | 2, 7 | sylbi 216 | . . 3 ⊢ (𝑥 ∈ (𝐴 +ℋ 𝐴) → 𝑥 ∈ 𝐴) |
9 | 8 | ssriv 3980 | . 2 ⊢ (𝐴 +ℋ 𝐴) ⊆ 𝐴 |
10 | 1, 1 | shsub1i 31254 | . 2 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐴) |
11 | 9, 10 | eqssi 3993 | 1 ⊢ (𝐴 +ℋ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 (class class class)co 7419 +ℎ cva 30802 Sℋ csh 30810 +ℋ cph 30813 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-hilex 30881 ax-hfvadd 30882 ax-hvcom 30883 ax-hvass 30884 ax-hv0cl 30885 ax-hvaddid 30886 ax-hfvmul 30887 ax-hvmulid 30888 ax-hvdistr2 30891 ax-hvmul0 30892 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-sub 11478 df-neg 11479 df-grpo 30375 df-ablo 30427 df-hvsub 30853 df-sh 31089 df-shs 31190 |
This theorem is referenced by: shslubi 31267 |
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