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Mirrors > Home > HSE Home > Th. List > omlsilem | Structured version Visualization version GIF version |
Description: Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
omlsilem.1 | ⊢ 𝐺 ∈ Sℋ |
omlsilem.2 | ⊢ 𝐻 ∈ Sℋ |
omlsilem.3 | ⊢ 𝐺 ⊆ 𝐻 |
omlsilem.4 | ⊢ (𝐻 ∩ (⊥‘𝐺)) = 0ℋ |
omlsilem.5 | ⊢ 𝐴 ∈ 𝐻 |
omlsilem.6 | ⊢ 𝐵 ∈ 𝐺 |
omlsilem.7 | ⊢ 𝐶 ∈ (⊥‘𝐺) |
Ref | Expression |
---|---|
omlsilem | ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐴 ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omlsilem.2 | . . . . . . . . . 10 ⊢ 𝐻 ∈ Sℋ | |
2 | omlsilem.5 | . . . . . . . . . 10 ⊢ 𝐴 ∈ 𝐻 | |
3 | 1, 2 | shelii 28644 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℋ |
4 | omlsilem.1 | . . . . . . . . . 10 ⊢ 𝐺 ∈ Sℋ | |
5 | omlsilem.6 | . . . . . . . . . 10 ⊢ 𝐵 ∈ 𝐺 | |
6 | 4, 5 | shelii 28644 | . . . . . . . . 9 ⊢ 𝐵 ∈ ℋ |
7 | shocss 28717 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Sℋ → (⊥‘𝐺) ⊆ ℋ) | |
8 | 4, 7 | ax-mp 5 | . . . . . . . . . 10 ⊢ (⊥‘𝐺) ⊆ ℋ |
9 | omlsilem.7 | . . . . . . . . . 10 ⊢ 𝐶 ∈ (⊥‘𝐺) | |
10 | 8, 9 | sselii 3818 | . . . . . . . . 9 ⊢ 𝐶 ∈ ℋ |
11 | 3, 6, 10 | hvsubaddi 28495 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
12 | eqcom 2785 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐶) = 𝐴 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) | |
13 | 11, 12 | bitri 267 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) |
14 | omlsilem.3 | . . . . . . . . . 10 ⊢ 𝐺 ⊆ 𝐻 | |
15 | 14, 5 | sselii 3818 | . . . . . . . . 9 ⊢ 𝐵 ∈ 𝐻 |
16 | shsubcl 28649 | . . . . . . . . 9 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻) | |
17 | 1, 2, 15, 16 | mp3an 1534 | . . . . . . . 8 ⊢ (𝐴 −ℎ 𝐵) ∈ 𝐻 |
18 | eleq1 2847 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → ((𝐴 −ℎ 𝐵) ∈ 𝐻 ↔ 𝐶 ∈ 𝐻)) | |
19 | 17, 18 | mpbii 225 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → 𝐶 ∈ 𝐻) |
20 | 13, 19 | sylbir 227 | . . . . . 6 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 ∈ 𝐻) |
21 | omlsilem.4 | . . . . . . . 8 ⊢ (𝐻 ∩ (⊥‘𝐺)) = 0ℋ | |
22 | 21 | eleq2i 2851 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ 𝐶 ∈ 0ℋ) |
23 | elin 4019 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ (𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺))) | |
24 | elch0 28683 | . . . . . . 7 ⊢ (𝐶 ∈ 0ℋ ↔ 𝐶 = 0ℎ) | |
25 | 22, 23, 24 | 3bitr3i 293 | . . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺)) ↔ 𝐶 = 0ℎ) |
26 | 20, 9, 25 | sylanblc 583 | . . . . 5 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 = 0ℎ) |
27 | 26 | oveq2d 6938 | . . . 4 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = (𝐵 +ℎ 0ℎ)) |
28 | ax-hvaddid 28433 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 +ℎ 0ℎ) = 𝐵) | |
29 | 6, 28 | ax-mp 5 | . . . 4 ⊢ (𝐵 +ℎ 0ℎ) = 𝐵 |
30 | 27, 29 | syl6eq 2830 | . . 3 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = 𝐵) |
31 | 30, 5 | syl6eqel 2867 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) ∈ 𝐺) |
32 | eleq1 2847 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐴 ∈ 𝐺 ↔ (𝐵 +ℎ 𝐶) ∈ 𝐺)) | |
33 | 31, 32 | mpbird 249 | 1 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ⊆ wss 3792 ‘cfv 6135 (class class class)co 6922 ℋchba 28348 +ℎ cva 28349 0ℎc0v 28353 −ℎ cmv 28354 Sℋ csh 28357 ⊥cort 28359 0ℋc0h 28364 |
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 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-hilex 28428 ax-hfvadd 28429 ax-hvcom 28430 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his2 28512 ax-his3 28513 |
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-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 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-neg 10609 df-hvsub 28400 df-sh 28636 df-oc 28681 df-ch0 28682 |
This theorem is referenced by: omlsii 28834 |
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