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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 31396 | . . . 4 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐶 +ℋ 𝐵)) |
| 5 | 2, 3 | shscomi 31382 | . . . 4 ⊢ (𝐶 +ℋ 𝐵) = (𝐵 +ℋ 𝐶) |
| 6 | 4, 5 | sseqtrdi 4024 | . . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 +ℋ 𝐵) ⊆ (𝐵 +ℋ 𝐶)) |
| 7 | 3, 2, 2 | shlessi 31396 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ (𝐶 +ℋ 𝐶)) |
| 8 | 2 | shsidmi 31403 | . . . 4 ⊢ (𝐶 +ℋ 𝐶) = 𝐶 |
| 9 | 7, 8 | sseqtrdi 4024 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 +ℋ 𝐶) ⊆ 𝐶) |
| 10 | 6, 9 | sylan9ss 3997 | . 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 +ℋ 𝐵) ⊆ 𝐶) |
| 11 | 1, 3 | shsub1i 31391 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 +ℋ 𝐵) |
| 12 | sstr 3992 | . . . 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 31392 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 +ℋ 𝐵) |
| 15 | sstr 3992 | . . . 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 2108 ⊆ wss 3951 (class class class)co 7431 Sℋ csh 30947 +ℋ cph 30950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvdistr2 31028 ax-hvmul0 31029 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-grpo 30512 df-ablo 30564 df-hvsub 30990 df-sh 31226 df-shs 31327 |
| This theorem is referenced by: shlesb1i 31405 shsval2i 31406 |
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