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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 28622 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
3 | ax-hvaddid 28412 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ 0ℎ) = 𝑥) | |
4 | 3 | eqcomd 2831 | . . . . . 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 28624 | . . . . . . 7 ⊢ (𝐵 ∈ Sℋ → 0ℎ ∈ 𝐵) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐵 |
9 | rspceov 6956 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 0ℎ ∈ 𝐵 ∧ 𝑥 = (𝑥 +ℎ 0ℎ)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) | |
10 | 8, 9 | mp3an2 1577 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = (𝑥 +ℎ 0ℎ)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
11 | 5, 10 | mpdan 678 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
12 | 6 | sheli 28622 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ) |
13 | hvaddid2 28431 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
14 | 13 | eqcomd 2831 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → 𝑥 = (0ℎ +ℎ 𝑥)) |
15 | 12, 14 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0ℎ +ℎ 𝑥)) |
16 | sh0 28624 | . . . . . . 7 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
17 | 1, 16 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐴 |
18 | rspceov 6956 | . . . . . 6 ⊢ ((0ℎ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 = (0ℎ +ℎ 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) | |
19 | 17, 18 | mp3an1 1576 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = (0ℎ +ℎ 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
20 | 15, 19 | mpdan 678 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
21 | 11, 20 | jaoi 888 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
22 | elun 3982 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
23 | 1, 6 | shseli 28726 | . . 3 ⊢ (𝑥 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
24 | 21, 22, 23 | 3imtr4i 284 | . 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ (𝐴 +ℋ 𝐵)) |
25 | 24 | ssriv 3831 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 +ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 878 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ∪ cun 3796 ⊆ wss 3798 (class class class)co 6910 ℋchba 28327 +ℎ cva 28328 0ℎc0v 28332 Sℋ csh 28336 +ℋ cph 28339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-hilex 28407 ax-hfvadd 28408 ax-hvcom 28409 ax-hvass 28410 ax-hv0cl 28411 ax-hvaddid 28412 ax-hfvmul 28413 ax-hvmulid 28414 ax-hvdistr2 28417 ax-hvmul0 28418 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-sub 10594 df-neg 10595 df-grpo 27899 df-ablo 27951 df-hvsub 28379 df-sh 28615 df-shs 28718 |
This theorem is referenced by: shsval2i 28797 shjshsi 28902 spanuni 28954 |
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