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Mirrors > Home > HSE Home > Th. List > shsva | Structured version Visualization version GIF version |
Description: Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shsva | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2730 | . . 3 ⊢ (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝐷) | |
2 | rspceov 7460 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑥 +ℎ 𝑦)) | |
3 | 1, 2 | mp3an3 1448 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑥 +ℎ 𝑦)) |
4 | shsel 30832 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ((𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑥 +ℎ 𝑦))) | |
5 | 3, 4 | imbitrrid 245 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 (class class class)co 7413 +ℎ cva 30438 Sℋ csh 30446 +ℋ cph 30449 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-hilex 30517 ax-hfvadd 30518 ax-hvcom 30519 ax-hvass 30520 ax-hv0cl 30521 ax-hvaddid 30522 ax-hfvmul 30523 ax-hvmulid 30524 ax-hvdistr2 30527 ax-hvmul0 30528 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-ltxr 11259 df-sub 11452 df-neg 11453 df-grpo 30011 df-ablo 30063 df-hvsub 30489 df-sh 30725 df-shs 30826 |
This theorem is referenced by: shsel1 30839 shsvai 30882 spanpr 31098 chscllem4 31158 |
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