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Mirrors > Home > HSE Home > Th. List > shsel | Structured version Visualization version GIF version |
Description: Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shsel | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shsval 27555 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = ( +ℎ “ (𝐴 × 𝐵))) | |
2 | 1 | eleq2d 2673 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ 𝐶 ∈ ( +ℎ “ (𝐴 × 𝐵)))) |
3 | ax-hfvadd 27241 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
4 | ffn 5958 | . . . 4 ⊢ ( +ℎ :( ℋ × ℋ)⟶ ℋ → +ℎ Fn ( ℋ × ℋ)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ +ℎ Fn ( ℋ × ℋ) |
6 | shss 27451 | . . . 4 ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ) | |
7 | shss 27451 | . . . 4 ⊢ (𝐵 ∈ Sℋ → 𝐵 ⊆ ℋ) | |
8 | xpss12 5148 | . . . 4 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 × 𝐵) ⊆ ( ℋ × ℋ)) | |
9 | 6, 7, 8 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 × 𝐵) ⊆ ( ℋ × ℋ)) |
10 | ovelimab 6710 | . . 3 ⊢ (( +ℎ Fn ( ℋ × ℋ) ∧ (𝐴 × 𝐵) ⊆ ( ℋ × ℋ)) → (𝐶 ∈ ( +ℎ “ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))) | |
11 | 5, 9, 10 | sylancr 694 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ ( +ℎ “ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))) |
12 | 2, 11 | bitrd 267 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 × cxp 5036 “ cima 5041 Fn wfn 5799 ⟶wf 5800 (class class class)co 6549 ℋchil 27160 +ℎ cva 27161 Sℋ csh 27169 +ℋ cph 27172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvdistr2 27250 ax-hvmul0 27251 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-grpo 26731 df-ablo 26783 df-hvsub 27212 df-sh 27448 df-shs 27551 |
This theorem is referenced by: shsel3 27558 shseli 27559 shscom 27562 shsva 27563 shless 27602 pjhth 27636 pjhtheu 27637 pjpreeq 27641 pjpjpre 27662 chscllem4 27883 sumdmdii 28658 sumdmdlem 28661 |
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