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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 28992 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℋ |
4 | omlsilem.1 | . . . . . . . . . 10 ⊢ 𝐺 ∈ Sℋ | |
5 | omlsilem.6 | . . . . . . . . . 10 ⊢ 𝐵 ∈ 𝐺 | |
6 | 4, 5 | shelii 28992 | . . . . . . . . 9 ⊢ 𝐵 ∈ ℋ |
7 | shocss 29063 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Sℋ → (⊥‘𝐺) ⊆ ℋ) | |
8 | 4, 7 | ax-mp 5 | . . . . . . . . . 10 ⊢ (⊥‘𝐺) ⊆ ℋ |
9 | omlsilem.7 | . . . . . . . . . 10 ⊢ 𝐶 ∈ (⊥‘𝐺) | |
10 | 8, 9 | sselii 3964 | . . . . . . . . 9 ⊢ 𝐶 ∈ ℋ |
11 | 3, 6, 10 | hvsubaddi 28843 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
12 | eqcom 2828 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐶) = 𝐴 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) | |
13 | 11, 12 | bitri 277 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) |
14 | omlsilem.3 | . . . . . . . . . 10 ⊢ 𝐺 ⊆ 𝐻 | |
15 | 14, 5 | sselii 3964 | . . . . . . . . 9 ⊢ 𝐵 ∈ 𝐻 |
16 | shsubcl 28997 | . . . . . . . . 9 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻) | |
17 | 1, 2, 15, 16 | mp3an 1457 | . . . . . . . 8 ⊢ (𝐴 −ℎ 𝐵) ∈ 𝐻 |
18 | eleq1 2900 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → ((𝐴 −ℎ 𝐵) ∈ 𝐻 ↔ 𝐶 ∈ 𝐻)) | |
19 | 17, 18 | mpbii 235 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → 𝐶 ∈ 𝐻) |
20 | 13, 19 | sylbir 237 | . . . . . 6 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 ∈ 𝐻) |
21 | omlsilem.4 | . . . . . . . 8 ⊢ (𝐻 ∩ (⊥‘𝐺)) = 0ℋ | |
22 | 21 | eleq2i 2904 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ 𝐶 ∈ 0ℋ) |
23 | elin 4169 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ (𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺))) | |
24 | elch0 29031 | . . . . . . 7 ⊢ (𝐶 ∈ 0ℋ ↔ 𝐶 = 0ℎ) | |
25 | 22, 23, 24 | 3bitr3i 303 | . . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺)) ↔ 𝐶 = 0ℎ) |
26 | 20, 9, 25 | sylanblc 591 | . . . . 5 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 = 0ℎ) |
27 | 26 | oveq2d 7172 | . . . 4 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = (𝐵 +ℎ 0ℎ)) |
28 | ax-hvaddid 28781 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 +ℎ 0ℎ) = 𝐵) | |
29 | 6, 28 | ax-mp 5 | . . . 4 ⊢ (𝐵 +ℎ 0ℎ) = 𝐵 |
30 | 27, 29 | syl6eq 2872 | . . 3 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = 𝐵) |
31 | 30, 5 | eqeltrdi 2921 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) ∈ 𝐺) |
32 | eleq1 2900 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐴 ∈ 𝐺 ↔ (𝐵 +ℎ 𝐶) ∈ 𝐺)) | |
33 | 31, 32 | mpbird 259 | 1 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐴 ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 ℋchba 28696 +ℎ cva 28697 0ℎc0v 28701 −ℎ cmv 28702 Sℋ csh 28705 ⊥cort 28707 0ℋc0h 28712 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvdistr2 28786 ax-hvmul0 28787 ax-hfi 28856 ax-his2 28860 ax-his3 28861 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-hvsub 28748 df-sh 28984 df-oc 29029 df-ch0 29030 |
This theorem is referenced by: omlsii 29180 |
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