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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 31234 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℋ | 
| 4 | omlsilem.1 | . . . . . . . . . 10 ⊢ 𝐺 ∈ Sℋ | |
| 5 | omlsilem.6 | . . . . . . . . . 10 ⊢ 𝐵 ∈ 𝐺 | |
| 6 | 4, 5 | shelii 31234 | . . . . . . . . 9 ⊢ 𝐵 ∈ ℋ | 
| 7 | shocss 31305 | . . . . . . . . . . 11 ⊢ (𝐺 ∈ Sℋ → (⊥‘𝐺) ⊆ ℋ) | |
| 8 | 4, 7 | ax-mp 5 | . . . . . . . . . 10 ⊢ (⊥‘𝐺) ⊆ ℋ | 
| 9 | omlsilem.7 | . . . . . . . . . 10 ⊢ 𝐶 ∈ (⊥‘𝐺) | |
| 10 | 8, 9 | sselii 3980 | . . . . . . . . 9 ⊢ 𝐶 ∈ ℋ | 
| 11 | 3, 6, 10 | hvsubaddi 31085 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) | 
| 12 | eqcom 2744 | . . . . . . . 8 ⊢ ((𝐵 +ℎ 𝐶) = 𝐴 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) | |
| 13 | 11, 12 | bitri 275 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ 𝐴 = (𝐵 +ℎ 𝐶)) | 
| 14 | omlsilem.3 | . . . . . . . . . 10 ⊢ 𝐺 ⊆ 𝐻 | |
| 15 | 14, 5 | sselii 3980 | . . . . . . . . 9 ⊢ 𝐵 ∈ 𝐻 | 
| 16 | shsubcl 31239 | . . . . . . . . 9 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻) | |
| 17 | 1, 2, 15, 16 | mp3an 1463 | . . . . . . . 8 ⊢ (𝐴 −ℎ 𝐵) ∈ 𝐻 | 
| 18 | eleq1 2829 | . . . . . . . 8 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → ((𝐴 −ℎ 𝐵) ∈ 𝐻 ↔ 𝐶 ∈ 𝐻)) | |
| 19 | 17, 18 | mpbii 233 | . . . . . . 7 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 → 𝐶 ∈ 𝐻) | 
| 20 | 13, 19 | sylbir 235 | . . . . . 6 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 ∈ 𝐻) | 
| 21 | omlsilem.4 | . . . . . . . 8 ⊢ (𝐻 ∩ (⊥‘𝐺)) = 0ℋ | |
| 22 | 21 | eleq2i 2833 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ 𝐶 ∈ 0ℋ) | 
| 23 | elin 3967 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ (𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺))) | |
| 24 | elch0 31273 | . . . . . . 7 ⊢ (𝐶 ∈ 0ℋ ↔ 𝐶 = 0ℎ) | |
| 25 | 22, 23, 24 | 3bitr3i 301 | . . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺)) ↔ 𝐶 = 0ℎ) | 
| 26 | 20, 9, 25 | sylanblc 589 | . . . . 5 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 = 0ℎ) | 
| 27 | 26 | oveq2d 7447 | . . . 4 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = (𝐵 +ℎ 0ℎ)) | 
| 28 | ax-hvaddid 31023 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 +ℎ 0ℎ) = 𝐵) | |
| 29 | 6, 28 | ax-mp 5 | . . . 4 ⊢ (𝐵 +ℎ 0ℎ) = 𝐵 | 
| 30 | 27, 29 | eqtrdi 2793 | . . 3 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = 𝐵) | 
| 31 | 30, 5 | eqeltrdi 2849 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) ∈ 𝐺) | 
| 32 | eleq1 2829 | . 2 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐴 ∈ 𝐺 ↔ (𝐵 +ℎ 𝐶) ∈ 𝐺)) | |
| 33 | 31, 32 | mpbird 257 | 1 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐴 ∈ 𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 ℋchba 30938 +ℎ cva 30939 0ℎc0v 30943 −ℎ cmv 30944 Sℋ csh 30947 ⊥cort 30949 0ℋc0h 30954 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his2 31102 ax-his3 31103 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-hvsub 30990 df-sh 31226 df-oc 31271 df-ch0 31272 | 
| This theorem is referenced by: omlsii 31422 | 
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