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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 3909 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
2 | 1 | biantrur 534 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵)) |
3 | shlesb1.2 | . . 3 ⊢ 𝐵 ∈ Sℋ | |
4 | shlesb1.1 | . . 3 ⊢ 𝐴 ∈ Sℋ | |
5 | 3, 4, 3 | shslubi 29420 | . 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) ↔ (𝐵 +ℋ 𝐴) ⊆ 𝐵) |
6 | 3, 4 | shsub2i 29408 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
7 | eqss 3902 | . . . 4 ⊢ ((𝐴 +ℋ 𝐵) = 𝐵 ↔ ((𝐴 +ℋ 𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 +ℋ 𝐵))) | |
8 | 6, 7 | mpbiran2 710 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) = 𝐵 ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐵) |
9 | 4, 3 | shscomi 29398 | . . . 4 ⊢ (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴) |
10 | 9 | sseq1i 3915 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐵 ↔ (𝐵 +ℋ 𝐴) ⊆ 𝐵) |
11 | 8, 10 | bitr2i 279 | . 2 ⊢ ((𝐵 +ℋ 𝐴) ⊆ 𝐵 ↔ (𝐴 +ℋ 𝐵) = 𝐵) |
12 | 2, 5, 11 | 3bitri 300 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 +ℋ 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 (class class class)co 7191 Sℋ csh 28963 +ℋ cph 28966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-hilex 29034 ax-hfvadd 29035 ax-hvcom 29036 ax-hvass 29037 ax-hv0cl 29038 ax-hvaddid 29039 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvdistr2 29044 ax-hvmul0 29045 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-neg 11030 df-grpo 28528 df-ablo 28580 df-hvsub 29006 df-sh 29242 df-shs 29343 |
This theorem is referenced by: shmodsi 29424 |
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