Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > shunssi | Structured version Visualization version GIF version |
Description: Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shincl.1 | ⊢ 𝐴 ∈ Sℋ |
shincl.2 | ⊢ 𝐵 ∈ Sℋ |
Ref | Expression |
---|---|
shunssi | ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 +ℋ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shincl.1 | . . . . . . 7 ⊢ 𝐴 ∈ Sℋ | |
2 | 1 | sheli 28991 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
3 | ax-hvaddid 28781 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ 0ℎ) = 𝑥) | |
4 | 3 | eqcomd 2827 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → 𝑥 = (𝑥 +ℎ 0ℎ)) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 = (𝑥 +ℎ 0ℎ)) |
6 | shincl.2 | . . . . . . 7 ⊢ 𝐵 ∈ Sℋ | |
7 | sh0 28993 | . . . . . . 7 ⊢ (𝐵 ∈ Sℋ → 0ℎ ∈ 𝐵) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐵 |
9 | rspceov 7203 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 0ℎ ∈ 𝐵 ∧ 𝑥 = (𝑥 +ℎ 0ℎ)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) | |
10 | 8, 9 | mp3an2 1445 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = (𝑥 +ℎ 0ℎ)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
11 | 5, 10 | mpdan 685 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
12 | 6 | sheli 28991 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ) |
13 | hvaddid2 28800 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
14 | 13 | eqcomd 2827 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → 𝑥 = (0ℎ +ℎ 𝑥)) |
15 | 12, 14 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0ℎ +ℎ 𝑥)) |
16 | sh0 28993 | . . . . . . 7 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
17 | 1, 16 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐴 |
18 | rspceov 7203 | . . . . . 6 ⊢ ((0ℎ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 = (0ℎ +ℎ 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) | |
19 | 17, 18 | mp3an1 1444 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = (0ℎ +ℎ 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
20 | 15, 19 | mpdan 685 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
21 | 11, 20 | jaoi 853 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
22 | elun 4125 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
23 | 1, 6 | shseli 29093 | . . 3 ⊢ (𝑥 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
24 | 21, 22, 23 | 3imtr4i 294 | . 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ (𝐴 +ℋ 𝐵)) |
25 | 24 | ssriv 3971 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 +ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ∪ cun 3934 ⊆ wss 3936 (class class class)co 7156 ℋchba 28696 +ℎ cva 28697 0ℎc0v 28701 Sℋ csh 28705 +ℋ cph 28708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvdistr2 28786 ax-hvmul0 28787 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-grpo 28270 df-ablo 28322 df-hvsub 28748 df-sh 28984 df-shs 29085 |
This theorem is referenced by: shsval2i 29164 shjshsi 29269 spanuni 29321 |
Copyright terms: Public domain | W3C validator |