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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 28666 | . . . . . . 7 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐶 ∈ 𝐻) → (𝐴 −ℎ 𝐶) ∈ 𝐻) | |
5 | 1, 2, 3, 4 | syl3anc 1439 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐻) |
6 | shuni.8 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷)) | |
7 | shel 28657 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) | |
8 | 1, 2, 7 | syl2anc 579 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℋ) |
9 | shuni.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Sℋ ) | |
10 | shuni.5 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
11 | shel 28657 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐵 ∈ 𝐾) → 𝐵 ∈ ℋ) | |
12 | 9, 10, 11 | syl2anc 579 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℋ) |
13 | shel 28657 | . . . . . . . . . 10 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐶 ∈ 𝐻) → 𝐶 ∈ ℋ) | |
14 | 1, 3, 13 | syl2anc 579 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℋ) |
15 | shuni.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝐾) | |
16 | shel 28657 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾) → 𝐷 ∈ ℋ) | |
17 | 9, 15, 16 | syl2anc 579 | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℋ) |
18 | hvaddsub4 28524 | . . . . . . . . 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 829 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷) ↔ (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵))) |
20 | 6, 19 | mpbid 224 | . . . . . . 7 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵)) |
21 | shsubcl 28666 | . . . . . . . 8 ⊢ ((𝐾 ∈ Sℋ ∧ 𝐷 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐷 −ℎ 𝐵) ∈ 𝐾) | |
22 | 9, 15, 10, 21 | syl3anc 1439 | . . . . . . 7 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) ∈ 𝐾) |
23 | 20, 22 | eqeltrd 2859 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 𝐾) |
24 | 5, 23 | elind 4021 | . . . . 5 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ (𝐻 ∩ 𝐾)) |
25 | shuni.3 | . . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐾) = 0ℋ) | |
26 | 24, 25 | eleqtrd 2861 | . . . 4 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) ∈ 0ℋ) |
27 | elch0 28700 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) ∈ 0ℋ ↔ (𝐴 −ℎ 𝐶) = 0ℎ) | |
28 | 26, 27 | sylib 210 | . . 3 ⊢ (𝜑 → (𝐴 −ℎ 𝐶) = 0ℎ) |
29 | hvsubeq0 28514 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) | |
30 | 8, 14, 29 | syl2anc 579 | . . 3 ⊢ (𝜑 → ((𝐴 −ℎ 𝐶) = 0ℎ ↔ 𝐴 = 𝐶)) |
31 | 28, 30 | mpbid 224 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
32 | 20, 28 | eqtr3d 2816 | . . . 4 ⊢ (𝜑 → (𝐷 −ℎ 𝐵) = 0ℎ) |
33 | hvsubeq0 28514 | . . . . 5 ⊢ ((𝐷 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) | |
34 | 17, 12, 33 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ((𝐷 −ℎ 𝐵) = 0ℎ ↔ 𝐷 = 𝐵)) |
35 | 32, 34 | mpbid 224 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) |
36 | 35 | eqcomd 2784 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) |
37 | 31, 36 | jca 507 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 (class class class)co 6924 ℋchba 28365 +ℎ cva 28366 0ℎc0v 28370 −ℎ cmv 28371 Sℋ csh 28374 0ℋc0h 28381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-hilex 28445 ax-hfvadd 28446 ax-hvcom 28447 ax-hvass 28448 ax-hv0cl 28449 ax-hvaddid 28450 ax-hfvmul 28451 ax-hvmulid 28452 ax-hvmulass 28453 ax-hvdistr1 28454 ax-hvdistr2 28455 ax-hvmul0 28456 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-hvsub 28417 df-sh 28653 df-ch0 28699 |
This theorem is referenced by: chocunii 28749 pjhthmo 28750 chscllem3 29087 |
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