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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 29148 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐶 +ℋ 𝐵)) |
5 | 2, 3 | shscomi 29134 | . . . 4 ⊢ (𝐶 +ℋ 𝐵) = (𝐵 +ℋ 𝐶) |
6 | 4, 5 | sseqtrdi 4016 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐵 +ℋ 𝐶)) |
7 | 3, 2, 2 | shlessi 29148 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ (𝐶 +ℋ 𝐶)) |
8 | 2 | shsidmi 29155 | . . . 4 ⊢ (𝐶 +ℋ 𝐶) = 𝐶 |
9 | 7, 8 | sseqtrdi 4016 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ 𝐶) |
10 | 6, 9 | sylan9ss 3979 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
11 | 1, 3 | shsub1i 29143 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵) |
12 | sstr 3974 | . . . 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 29144 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
15 | sstr 3974 | . . . 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 514 | . 2 ⊢ ((𝐴 +ℋ 𝐵) ⊆ 𝐶 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
18 | 10, 17 | impbii 211 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ⊆ wss 3935 (class class class)co 7150 Sℋ csh 28699 +ℋ cph 28702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvdistr2 28780 ax-hvmul0 28781 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-grpo 28264 df-ablo 28316 df-hvsub 28742 df-sh 28978 df-shs 29079 |
This theorem is referenced by: shlesb1i 29157 shsval2i 29158 |
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