Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 29605 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℋ |
4 | omlsilem.1 | . . . . . . . . . 10 ⊢ 𝐺 ∈ Sℋ | |
5 | omlsilem.6 | . . . . . . . . . 10 ⊢ 𝐵 ∈ 𝐺 | |
6 | 4, 5 | shelii 29605 | . . . . . . . . 9 ⊢ 𝐵 ∈ ℋ |
7 | shocss 29676 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Sℋ → (⊥‘𝐺) ⊆ ℋ) | |
8 | 4, 7 | ax-mp 5 | . . . . . . . . . 10 ⊢ (⊥‘𝐺) ⊆ ℋ |
9 | omlsilem.7 | . . . . . . . . . 10 ⊢ 𝐶 ∈ (⊥‘𝐺) | |
10 | 8, 9 | sselii 3920 | . . . . . . . . 9 ⊢ 𝐶 ∈ ℋ |
11 | 3, 6, 10 | hvsubaddi 29456 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
12 | eqcom 2740 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐶) = 𝐴 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) | |
13 | 11, 12 | bitri 274 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) |
14 | omlsilem.3 | . . . . . . . . . 10 ⊢ 𝐺 ⊆ 𝐻 | |
15 | 14, 5 | sselii 3920 | . . . . . . . . 9 ⊢ 𝐵 ∈ 𝐻 |
16 | shsubcl 29610 | . . . . . . . . 9 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻) | |
17 | 1, 2, 15, 16 | mp3an 1459 | . . . . . . . 8 ⊢ (𝐴 −ℎ 𝐵) ∈ 𝐻 |
18 | eleq1 2821 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → ((𝐴 −ℎ 𝐵) ∈ 𝐻 ↔ 𝐶 ∈ 𝐻)) | |
19 | 17, 18 | mpbii 232 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → 𝐶 ∈ 𝐻) |
20 | 13, 19 | sylbir 234 | . . . . . 6 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 ∈ 𝐻) |
21 | omlsilem.4 | . . . . . . . 8 ⊢ (𝐻 ∩ (⊥‘𝐺)) = 0ℋ | |
22 | 21 | eleq2i 2825 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ 𝐶 ∈ 0ℋ) |
23 | elin 3905 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ (𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺))) | |
24 | elch0 29644 | . . . . . . 7 ⊢ (𝐶 ∈ 0ℋ ↔ 𝐶 = 0ℎ) | |
25 | 22, 23, 24 | 3bitr3i 300 | . . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺)) ↔ 𝐶 = 0ℎ) |
26 | 20, 9, 25 | sylanblc 588 | . . . . 5 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 = 0ℎ) |
27 | 26 | oveq2d 7311 | . . . 4 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = (𝐵 +ℎ 0ℎ)) |
28 | ax-hvaddid 29394 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 +ℎ 0ℎ) = 𝐵) | |
29 | 6, 28 | ax-mp 5 | . . . 4 ⊢ (𝐵 +ℎ 0ℎ) = 𝐵 |
30 | 27, 29 | eqtrdi 2789 | . . 3 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = 𝐵) |
31 | 30, 5 | eqeltrdi 2842 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) ∈ 𝐺) |
32 | eleq1 2821 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐴 ∈ 𝐺 ↔ (𝐵 +ℎ 𝐶) ∈ 𝐺)) | |
33 | 31, 32 | mpbird 256 | 1 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ∩ cin 3888 ⊆ wss 3889 ‘cfv 6447 (class class class)co 7295 ℋchba 29309 +ℎ cva 29310 0ℎc0v 29314 −ℎ cmv 29315 Sℋ csh 29318 ⊥cort 29320 0ℋc0h 29325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-hilex 29389 ax-hfvadd 29390 ax-hvcom 29391 ax-hvass 29392 ax-hv0cl 29393 ax-hvaddid 29394 ax-hfvmul 29395 ax-hvmulid 29396 ax-hvdistr2 29399 ax-hvmul0 29400 ax-hfi 29469 ax-his2 29473 ax-his3 29474 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-pnf 11039 df-mnf 11040 df-ltxr 11042 df-sub 11235 df-neg 11236 df-hvsub 29361 df-sh 29597 df-oc 29642 df-ch0 29643 |
This theorem is referenced by: omlsii 29793 |
Copyright terms: Public domain | W3C validator |