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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 29003 | . . . . . . 7 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐶 ∈ 𝐻) → (𝐴 −ℎ 𝐶) ∈ 𝐻) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐻) |
6 | shuni.8 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷)) | |
7 | shel 28994 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) | |
8 | 1, 2, 7 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℋ) |
9 | shuni.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Sℋ ) | |
10 | shuni.5 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
11 | shel 28994 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐵 ∈ 𝐾) → 𝐵 ∈ ℋ) | |
12 | 9, 10, 11 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℋ) |
13 | shel 28994 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐶 ∈ 𝐻) → 𝐶 ∈ ℋ) | |
14 | 1, 3, 13 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℋ) |
15 | shuni.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
16 | shel 28994 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾) → 𝐷 ∈ ℋ) | |
17 | 9, 15, 16 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℋ) |
18 | hvaddsub4 28861 | . . . . . . . . 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 837 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷) ↔ (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵))) |
20 | 6, 19 | mpbid 235 | . . . . . . 7 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵)) |
21 | shsubcl 29003 | . . . . . . . 8 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐷 −ℎ 𝐵) ∈ 𝐾) | |
22 | 9, 15, 10, 21 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) ∈ 𝐾) |
23 | 20, 22 | eqeltrd 2890 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐾) |
24 | 5, 23 | elind 4121 | . . . . 5 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ (𝐻 ∩ 𝐾)) |
25 | shuni.3 | . . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ) | |
26 | 24, 25 | eleqtrd 2892 | . . . 4 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 0ℋ) |
27 | elch0 29037 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) ∈ 0ℋ ↔ (𝐴 −ℎ 𝐶) = 0ℎ) | |
28 | 26, 27 | sylib 221 | . . 3 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = 0ℎ) |
29 | hvsubeq0 28851 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) | |
30 | 8, 14, 29 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) |
31 | 28, 30 | mpbid 235 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
32 | 20, 28 | eqtr3d 2835 | . . . 4 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) = 0ℎ) |
33 | hvsubeq0 28851 | . . . . 5 ⊢ ((𝐷 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) | |
34 | 17, 12, 33 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) |
35 | 32, 34 | mpbid 235 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) |
36 | 35 | eqcomd 2804 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) |
37 | 31, 36 | jca 515 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 (class class class)co 7135 ℋchba 28702 +ℎ cva 28703 0ℎc0v 28707 −ℎ cmv 28708 Sℋ csh 28711 0ℋc0h 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-hvsub 28754 df-sh 28990 df-ch0 29036 |
This theorem is referenced by: chocunii 29084 pjhthmo 29085 chscllem3 29422 |
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