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 28991 | . . . . . . 7 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐶 ∈ 𝐻) → (𝐴 −ℎ 𝐶) ∈ 𝐻) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐻) |
6 | shuni.8 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷)) | |
7 | shel 28982 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) | |
8 | 1, 2, 7 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℋ) |
9 | shuni.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Sℋ ) | |
10 | shuni.5 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
11 | shel 28982 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐵 ∈ 𝐾) → 𝐵 ∈ ℋ) | |
12 | 9, 10, 11 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℋ) |
13 | shel 28982 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐶 ∈ 𝐻) → 𝐶 ∈ ℋ) | |
14 | 1, 3, 13 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℋ) |
15 | shuni.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
16 | shel 28982 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾) → 𝐷 ∈ ℋ) | |
17 | 9, 15, 16 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℋ) |
18 | hvaddsub4 28849 | . . . . . . . . 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 836 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷) ↔ (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵))) |
20 | 6, 19 | mpbid 234 | . . . . . . 7 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵)) |
21 | shsubcl 28991 | . . . . . . . 8 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐷 −ℎ 𝐵) ∈ 𝐾) | |
22 | 9, 15, 10, 21 | syl3anc 1367 | . . . . . . 7 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) ∈ 𝐾) |
23 | 20, 22 | eqeltrd 2913 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐾) |
24 | 5, 23 | elind 4170 | . . . . 5 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ (𝐻 ∩ 𝐾)) |
25 | shuni.3 | . . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ) | |
26 | 24, 25 | eleqtrd 2915 | . . . 4 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 0ℋ) |
27 | elch0 29025 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) ∈ 0ℋ ↔ (𝐴 −ℎ 𝐶) = 0ℎ) | |
28 | 26, 27 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = 0ℎ) |
29 | hvsubeq0 28839 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) | |
30 | 8, 14, 29 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) |
31 | 28, 30 | mpbid 234 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
32 | 20, 28 | eqtr3d 2858 | . . . 4 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) = 0ℎ) |
33 | hvsubeq0 28839 | . . . . 5 ⊢ ((𝐷 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) | |
34 | 17, 12, 33 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) |
35 | 32, 34 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) |
36 | 35 | eqcomd 2827 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) |
37 | 31, 36 | jca 514 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 (class class class)co 7150 ℋchba 28690 +ℎ cva 28691 0ℎc0v 28695 −ℎ cmv 28696 Sℋ csh 28699 0ℋc0h 28706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-hvsub 28742 df-sh 28978 df-ch0 29024 |
This theorem is referenced by: chocunii 29072 pjhthmo 29073 chscllem3 29410 |
Copyright terms: Public domain | W3C validator |