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Mirrors > Home > HSE Home > Th. List > shslubi | Structured version Visualization version GIF version |
Description: The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shslub.1 | ⊢ 𝐴 ∈ Sℋ |
shslub.2 | ⊢ 𝐵 ∈ Sℋ |
shslub.3 | ⊢ 𝐶 ∈ Sℋ |
Ref | Expression |
---|---|
shslubi | ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shslub.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
2 | shslub.3 | . . . . 5 ⊢ 𝐶 ∈ Sℋ | |
3 | shslub.2 | . . . . 5 ⊢ 𝐵 ∈ Sℋ | |
4 | 1, 2, 3 | shlessi 31405 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐶 +ℋ 𝐵)) |
5 | 2, 3 | shscomi 31391 | . . . 4 ⊢ (𝐶 +ℋ 𝐵) = (𝐵 +ℋ 𝐶) |
6 | 4, 5 | sseqtrdi 4045 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐵 +ℋ 𝐶)) |
7 | 3, 2, 2 | shlessi 31405 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ (𝐶 +ℋ 𝐶)) |
8 | 2 | shsidmi 31412 | . . . 4 ⊢ (𝐶 +ℋ 𝐶) = 𝐶 |
9 | 7, 8 | sseqtrdi 4045 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ 𝐶) |
10 | 6, 9 | sylan9ss 4008 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
11 | 1, 3 | shsub1i 31400 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵) |
12 | sstr 4003 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 +ℋ 𝐵) ∧ (𝐴 +ℋ 𝐵) ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
13 | 11, 12 | mpan 690 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → 𝐴 ⊆ 𝐶) |
14 | 3, 1 | shsub2i 31401 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
15 | sstr 4003 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 +ℋ 𝐵) ∧ (𝐴 +ℋ 𝐵) ⊆ 𝐶) → 𝐵 ⊆ 𝐶) | |
16 | 14, 15 | mpan 690 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → 𝐵 ⊆ 𝐶) |
17 | 13, 16 | jca 511 | . 2 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
18 | 10, 17 | impbii 209 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2105 ⊆ wss 3962 (class class class)co 7430 Sℋ csh 30956 +ℋ cph 30959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-hilex 31027 ax-hfvadd 31028 ax-hvcom 31029 ax-hvass 31030 ax-hv0cl 31031 ax-hvaddid 31032 ax-hfvmul 31033 ax-hvmulid 31034 ax-hvdistr2 31037 ax-hvmul0 31038 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-sub 11491 df-neg 11492 df-grpo 30521 df-ablo 30573 df-hvsub 30999 df-sh 31235 df-shs 31336 |
This theorem is referenced by: shlesb1i 31414 shsval2i 31415 |
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