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Mirrors > Home > HSE Home > Th. List > shsss | Structured version Visualization version GIF version |
Description: The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shsss | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ⊆ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shsval 28772 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = ( +ℎ “ (𝐴 × 𝐵))) | |
2 | imassrn 5820 | . . 3 ⊢ ( +ℎ “ (𝐴 × 𝐵)) ⊆ ran +ℎ | |
3 | ax-hfvadd 28460 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
4 | frn 6391 | . . . 4 ⊢ ( +ℎ :( ℋ × ℋ)⟶ ℋ → ran +ℎ ⊆ ℋ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ran +ℎ ⊆ ℋ |
6 | 2, 5 | sstri 3900 | . 2 ⊢ ( +ℎ “ (𝐴 × 𝐵)) ⊆ ℋ |
7 | 1, 6 | syl6eqss 3944 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ⊆ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2080 ⊆ wss 3861 × cxp 5444 ran crn 5447 “ cima 5449 ⟶wf 6224 (class class class)co 7019 ℋchba 28379 +ℎ cva 28380 Sℋ csh 28388 +ℋ cph 28391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-hilex 28459 ax-hfvadd 28460 ax-hvcom 28461 ax-hvass 28462 ax-hv0cl 28463 ax-hvaddid 28464 ax-hfvmul 28465 ax-hvmulid 28466 ax-hvdistr2 28469 ax-hvmul0 28470 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-po 5365 df-so 5366 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-ltxr 10529 df-sub 10721 df-neg 10722 df-grpo 27953 df-ablo 28005 df-hvsub 28431 df-shs 28768 |
This theorem is referenced by: shscli 28777 pjhth 28853 pjpreeq 28858 |
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