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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 28999 | . . . . . . 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 28990 | . . . . . . . . . 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 28990 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐵 ∈ 𝐾) → 𝐵 ∈ ℋ) | |
12 | 9, 10, 11 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℋ) |
13 | shel 28990 | . . . . . . . . . 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 28990 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾) → 𝐷 ∈ ℋ) | |
17 | 9, 15, 16 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℋ) |
18 | hvaddsub4 28857 | . . . . . . . . 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 28999 | . . . . . . . 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 2915 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐾) |
24 | 5, 23 | elind 4173 | . . . . 5 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ (𝐻 ∩ 𝐾)) |
25 | shuni.3 | . . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ) | |
26 | 24, 25 | eleqtrd 2917 | . . . 4 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 0ℋ) |
27 | elch0 29033 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) ∈ 0ℋ ↔ (𝐴 −ℎ 𝐶) = 0ℎ) | |
28 | 26, 27 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = 0ℎ) |
29 | hvsubeq0 28847 | . . . 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 2860 | . . . 4 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) = 0ℎ) |
33 | hvsubeq0 28847 | . . . . 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 2829 | . 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 1537 ∈ wcel 2114 ∩ cin 3937 (class class class)co 7158 ℋchba 28698 +ℎ cva 28699 0ℎc0v 28703 −ℎ cmv 28704 Sℋ csh 28707 0ℋc0h 28714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-hvsub 28750 df-sh 28986 df-ch0 29032 |
This theorem is referenced by: chocunii 29080 pjhthmo 29081 chscllem3 29418 |
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