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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 31034 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = ( +ℎ “ (𝐴 × 𝐵))) | |
2 | imassrn 6060 | . . 3 ⊢ ( +ℎ “ (𝐴 × 𝐵)) ⊆ ran +ℎ | |
3 | ax-hfvadd 30722 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
4 | frn 6714 | . . . 4 ⊢ ( +ℎ :( ℋ × ℋ)⟶ ℋ → ran +ℎ ⊆ ℋ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ran +ℎ ⊆ ℋ |
6 | 2, 5 | sstri 3983 | . 2 ⊢ ( +ℎ “ (𝐴 × 𝐵)) ⊆ ℋ |
7 | 1, 6 | eqsstrdi 4028 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) ⊆ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ⊆ wss 3940 × cxp 5664 ran crn 5667 “ cima 5669 ⟶wf 6529 (class class class)co 7401 ℋchba 30641 +ℎ cva 30642 Sℋ csh 30650 +ℋ cph 30653 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-hilex 30721 ax-hfvadd 30722 ax-hvcom 30723 ax-hvass 30724 ax-hv0cl 30725 ax-hvaddid 30726 ax-hfvmul 30727 ax-hvmulid 30728 ax-hvdistr2 30731 ax-hvmul0 30732 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-ltxr 11250 df-sub 11443 df-neg 11444 df-grpo 30215 df-ablo 30267 df-hvsub 30693 df-shs 31030 |
This theorem is referenced by: shscli 31039 pjhth 31115 pjpreeq 31120 |
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