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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 30468 | . . . . . . . . 9 ⊢ 𝐴 ∈ ℋ |
4 | omlsilem.1 | . . . . . . . . . 10 ⊢ 𝐺 ∈ Sℋ | |
5 | omlsilem.6 | . . . . . . . . . 10 ⊢ 𝐵 ∈ 𝐺 | |
6 | 4, 5 | shelii 30468 | . . . . . . . . 9 ⊢ 𝐵 ∈ ℋ |
7 | shocss 30539 | . . . . . . . . . . 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 30319 | . . . . . . . 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 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 30473 | . . . . . . . . 9 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻) | |
17 | 1, 2, 15, 16 | mp3an 1462 | . . . . . . . 8 ⊢ (𝐴 −ℎ 𝐵) ∈ 𝐻 |
18 | eleq1 2822 | . . . . . . . 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 2826 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ 𝐶 ∈ 0ℋ) |
23 | elin 3965 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐻 ∩ (⊥‘𝐺)) ↔ (𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺))) | |
24 | elch0 30507 | . . . . . . 7 ⊢ (𝐶 ∈ 0ℋ ↔ 𝐶 = 0ℎ) | |
25 | 22, 23, 24 | 3bitr3i 301 | . . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐶 ∈ (⊥‘𝐺)) ↔ 𝐶 = 0ℎ) |
26 | 20, 9, 25 | sylanblc 590 | . . . . 5 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → 𝐶 = 0ℎ) |
27 | 26 | oveq2d 7425 | . . . 4 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) → (𝐵 +ℎ 𝐶) = (𝐵 +ℎ 0ℎ)) |
28 | ax-hvaddid 30257 | . . . . 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 2822 | . 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 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3948 ⊆ wss 3949 ‘cfv 6544 (class class class)co 7409 ℋchba 30172 +ℎ cva 30173 0ℎc0v 30177 −ℎ cmv 30178 Sℋ csh 30181 ⊥cort 30183 0ℋc0h 30188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his2 30336 ax-his3 30337 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 df-hvsub 30224 df-sh 30460 df-oc 30505 df-ch0 30506 |
This theorem is referenced by: omlsii 30656 |
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