![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > shuni | Structured version Visualization version GIF version |
Description: Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shuni.1 | ⊢ (𝜑 → 𝐻 ∈ Sℋ ) |
shuni.2 | ⊢ (𝜑 → 𝐾 ∈ Sℋ ) |
shuni.3 | ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ) |
shuni.4 | ⊢ (𝜑 → 𝐴 ∈ 𝐻) |
shuni.5 | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
shuni.6 | ⊢ (𝜑 → 𝐶 ∈ 𝐻) |
shuni.7 | ⊢ (𝜑 → 𝐷 ∈ 𝐾) |
shuni.8 | ⊢ (𝜑 → (𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷)) |
Ref | Expression |
---|---|
shuni | ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shuni.1 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Sℋ ) | |
2 | shuni.4 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐻) | |
3 | shuni.6 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐻) | |
4 | shsubcl 31029 | . . . . . . 7 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐶 ∈ 𝐻) → (𝐴 −ℎ 𝐶) ∈ 𝐻) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐻) |
6 | shuni.8 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷)) | |
7 | shel 31020 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) | |
8 | 1, 2, 7 | syl2anc 583 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℋ) |
9 | shuni.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Sℋ ) | |
10 | shuni.5 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
11 | shel 31020 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐵 ∈ 𝐾) → 𝐵 ∈ ℋ) | |
12 | 9, 10, 11 | syl2anc 583 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℋ) |
13 | shel 31020 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐶 ∈ 𝐻) → 𝐶 ∈ ℋ) | |
14 | 1, 3, 13 | syl2anc 583 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℋ) |
15 | shuni.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
16 | shel 31020 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾) → 𝐷 ∈ ℋ) | |
17 | 9, 15, 16 | syl2anc 583 | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℋ) |
18 | hvaddsub4 30887 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷) ↔ (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵))) | |
19 | 8, 12, 14, 17, 18 | syl22anc 838 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷) ↔ (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵))) |
20 | 6, 19 | mpbid 231 | . . . . . . 7 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵)) |
21 | shsubcl 31029 | . . . . . . . 8 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐷 −ℎ 𝐵) ∈ 𝐾) | |
22 | 9, 15, 10, 21 | syl3anc 1369 | . . . . . . 7 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) ∈ 𝐾) |
23 | 20, 22 | eqeltrd 2829 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐾) |
24 | 5, 23 | elind 4194 | . . . . 5 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ (𝐻 ∩ 𝐾)) |
25 | shuni.3 | . . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ) | |
26 | 24, 25 | eleqtrd 2831 | . . . 4 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 0ℋ) |
27 | elch0 31063 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) ∈ 0ℋ ↔ (𝐴 −ℎ 𝐶) = 0ℎ) | |
28 | 26, 27 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = 0ℎ) |
29 | hvsubeq0 30877 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) | |
30 | 8, 14, 29 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) |
31 | 28, 30 | mpbid 231 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
32 | 20, 28 | eqtr3d 2770 | . . . 4 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) = 0ℎ) |
33 | hvsubeq0 30877 | . . . . 5 ⊢ ((𝐷 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) | |
34 | 17, 12, 33 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) |
35 | 32, 34 | mpbid 231 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) |
36 | 35 | eqcomd 2734 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) |
37 | 31, 36 | jca 511 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 (class class class)co 7420 ℋchba 30728 +ℎ cva 30729 0ℎc0v 30733 −ℎ cmv 30734 Sℋ csh 30737 0ℋc0h 30744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-hilex 30808 ax-hfvadd 30809 ax-hvcom 30810 ax-hvass 30811 ax-hv0cl 30812 ax-hvaddid 30813 ax-hfvmul 30814 ax-hvmulid 30815 ax-hvmulass 30816 ax-hvdistr1 30817 ax-hvdistr2 30818 ax-hvmul0 30819 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 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 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-hvsub 30780 df-sh 31016 df-ch0 31062 |
This theorem is referenced by: chocunii 31110 pjhthmo 31111 chscllem3 31448 |
Copyright terms: Public domain | W3C validator |