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Mirrors > Home > HSE Home > Th. List > shseli | Structured version Visualization version GIF version |
Description: Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shscl.1 | ⊢ 𝐴 ∈ Sℋ |
shscl.2 | ⊢ 𝐵 ∈ Sℋ |
Ref | Expression |
---|---|
shseli | ⊢ (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shscl.1 | . 2 ⊢ 𝐴 ∈ Sℋ | |
2 | shscl.2 | . 2 ⊢ 𝐵 ∈ Sℋ | |
3 | shsel 29251 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐶 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 (class class class)co 7172 +ℎ cva 28857 Sℋ csh 28865 +ℋ cph 28868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-hilex 28936 ax-hfvadd 28937 ax-hvcom 28938 ax-hvass 28939 ax-hv0cl 28940 ax-hvaddid 28941 ax-hfvmul 28942 ax-hvmulid 28943 ax-hvdistr2 28946 ax-hvmul0 28947 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-pnf 10757 df-mnf 10758 df-ltxr 10760 df-sub 10952 df-neg 10953 df-grpo 28430 df-ablo 28482 df-hvsub 28908 df-sh 29144 df-shs 29245 |
This theorem is referenced by: shscli 29254 shunssi 29305 shsleji 29307 shsidmi 29321 shmodsi 29326 chseli 29396 spanuni 29481 spanunsni 29516 5oalem7 29597 pjjsi 29637 cdjreui 30369 cdj3lem2a 30373 cdj3lem2b 30374 cdj3lem3a 30376 |
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