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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 31207 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐶 +ℋ 𝐵)) |
5 | 2, 3 | shscomi 31193 | . . . 4 ⊢ (𝐶 +ℋ 𝐵) = (𝐵 +ℋ 𝐶) |
6 | 4, 5 | sseqtrdi 4032 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐵 +ℋ 𝐶)) |
7 | 3, 2, 2 | shlessi 31207 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ (𝐶 +ℋ 𝐶)) |
8 | 2 | shsidmi 31214 | . . . 4 ⊢ (𝐶 +ℋ 𝐶) = 𝐶 |
9 | 7, 8 | sseqtrdi 4032 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ 𝐶) |
10 | 6, 9 | sylan9ss 3995 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
11 | 1, 3 | shsub1i 31202 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵) |
12 | sstr 3990 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 +ℋ 𝐵) ∧ (𝐴 +ℋ 𝐵) ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
13 | 11, 12 | mpan 688 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → 𝐴 ⊆ 𝐶) |
14 | 3, 1 | shsub2i 31203 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
15 | sstr 3990 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 +ℋ 𝐵) ∧ (𝐴 +ℋ 𝐵) ⊆ 𝐶) → 𝐵 ⊆ 𝐶) | |
16 | 14, 15 | mpan 688 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → 𝐵 ⊆ 𝐶) |
17 | 13, 16 | jca 510 | . 2 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
18 | 10, 17 | impbii 208 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ⊆ wss 3949 (class class class)co 7426 Sℋ csh 30758 +ℋ cph 30761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-hilex 30829 ax-hfvadd 30830 ax-hvcom 30831 ax-hvass 30832 ax-hv0cl 30833 ax-hvaddid 30834 ax-hfvmul 30835 ax-hvmulid 30836 ax-hvdistr2 30839 ax-hvmul0 30840 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-sub 11484 df-neg 11485 df-grpo 30323 df-ablo 30375 df-hvsub 30801 df-sh 31037 df-shs 31138 |
This theorem is referenced by: shlesb1i 31216 shsval2i 31217 |
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