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Mirrors > Home > HSE Home > Th. List > shlesb1i | Structured version Visualization version GIF version |
Description: Hilbert lattice ordering in terms of subspace sum. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shlesb1.1 | ⊢ 𝐴 ∈ Sℋ |
shlesb1.2 | ⊢ 𝐵 ∈ Sℋ |
Ref | Expression |
---|---|
shlesb1i | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 +ℋ 𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3873 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
2 | 1 | biantrur 523 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵)) |
3 | shlesb1.2 | . . 3 ⊢ 𝐵 ∈ Sℋ | |
4 | shlesb1.1 | . . 3 ⊢ 𝐴 ∈ Sℋ | |
5 | 3, 4, 3 | shslubi 28937 | . 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) ↔ (𝐵 +ℋ 𝐴) ⊆ 𝐵) |
6 | 3, 4 | shsub2i 28925 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
7 | eqss 3867 | . . . 4 ⊢ ((𝐴 +ℋ 𝐵) = 𝐵 ↔ ((𝐴 +ℋ 𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 +ℋ 𝐵))) | |
8 | 6, 7 | mpbiran2 697 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) = 𝐵 ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐵) |
9 | 4, 3 | shscomi 28915 | . . . 4 ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴) |
10 | 9 | sseq1i 3879 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐵 ↔ (𝐵 +ℋ 𝐴) ⊆ 𝐵) |
11 | 8, 10 | bitr2i 268 | . 2 ⊢ ((𝐵 +ℋ 𝐴) ⊆ 𝐵 ↔ (𝐴 +ℋ 𝐵) = 𝐵) |
12 | 2, 5, 11 | 3bitri 289 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 +ℋ 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3823 (class class class)co 6970 Sℋ csh 28478 +ℋ cph 28481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-hilex 28549 ax-hfvadd 28550 ax-hvcom 28551 ax-hvass 28552 ax-hv0cl 28553 ax-hvaddid 28554 ax-hfvmul 28555 ax-hvmulid 28556 ax-hvdistr2 28559 ax-hvmul0 28560 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-po 5320 df-so 5321 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-pnf 10470 df-mnf 10471 df-ltxr 10473 df-sub 10666 df-neg 10667 df-grpo 28041 df-ablo 28093 df-hvsub 28521 df-sh 28757 df-shs 28860 |
This theorem is referenced by: shmodsi 28941 |
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