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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 31134 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐶 +ℋ 𝐵)) |
5 | 2, 3 | shscomi 31120 | . . . 4 ⊢ (𝐶 +ℋ 𝐵) = (𝐵 +ℋ 𝐶) |
6 | 4, 5 | sseqtrdi 4027 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐵 +ℋ 𝐶)) |
7 | 3, 2, 2 | shlessi 31134 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ (𝐶 +ℋ 𝐶)) |
8 | 2 | shsidmi 31141 | . . . 4 ⊢ (𝐶 +ℋ 𝐶) = 𝐶 |
9 | 7, 8 | sseqtrdi 4027 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ 𝐶) |
10 | 6, 9 | sylan9ss 3990 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
11 | 1, 3 | shsub1i 31129 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵) |
12 | sstr 3985 | . . . 4 ⊢ ((𝐴 ⊆ (𝐴 +ℋ 𝐵) ∧ (𝐴 +ℋ 𝐵) ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
13 | 11, 12 | mpan 687 | . . 3 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → 𝐴 ⊆ 𝐶) |
14 | 3, 1 | shsub2i 31130 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
15 | sstr 3985 | . . . 4 ⊢ ((𝐵 ⊆ (𝐴 +ℋ 𝐵) ∧ (𝐴 +ℋ 𝐵) ⊆ 𝐶) → 𝐵 ⊆ 𝐶) | |
16 | 14, 15 | mpan 687 | . . 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 208 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ⊆ wss 3943 (class class class)co 7404 Sℋ csh 30685 +ℋ cph 30688 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-hilex 30756 ax-hfvadd 30757 ax-hvcom 30758 ax-hvass 30759 ax-hv0cl 30760 ax-hvaddid 30761 ax-hfvmul 30762 ax-hvmulid 30763 ax-hvdistr2 30766 ax-hvmul0 30767 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-sub 11447 df-neg 11448 df-grpo 30250 df-ablo 30302 df-hvsub 30728 df-sh 30964 df-shs 31065 |
This theorem is referenced by: shlesb1i 31143 shsval2i 31144 |
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