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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 29563 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
3 | ax-hvaddid 29353 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ 0ℎ) = 𝑥) | |
4 | 3 | eqcomd 2744 | . . . . . 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 29565 | . . . . . . 7 ⊢ (𝐵 ∈ Sℋ → 0ℎ ∈ 𝐵) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐵 |
9 | rspceov 7316 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 0ℎ ∈ 𝐵 ∧ 𝑥 = (𝑥 +ℎ 0ℎ)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) | |
10 | 8, 9 | mp3an2 1448 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = (𝑥 +ℎ 0ℎ)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
11 | 5, 10 | mpdan 684 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
12 | 6 | sheli 29563 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ) |
13 | hvaddid2 29372 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
14 | 13 | eqcomd 2744 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → 𝑥 = (0ℎ +ℎ 𝑥)) |
15 | 12, 14 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0ℎ +ℎ 𝑥)) |
16 | sh0 29565 | . . . . . . 7 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
17 | 1, 16 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐴 |
18 | rspceov 7316 | . . . . . 6 ⊢ ((0ℎ ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 = (0ℎ +ℎ 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) | |
19 | 17, 18 | mp3an1 1447 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 = (0ℎ +ℎ 𝑥)) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
20 | 15, 19 | mpdan 684 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
21 | 11, 20 | jaoi 854 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
22 | elun 4084 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
23 | 1, 6 | shseli 29665 | . . 3 ⊢ (𝑥 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧)) |
24 | 21, 22, 23 | 3imtr4i 292 | . 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) → 𝑥 ∈ (𝐴 +ℋ 𝐵)) |
25 | 24 | ssriv 3926 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 +ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 ∪ cun 3886 ⊆ wss 3888 (class class class)co 7269 ℋchba 29268 +ℎ cva 29269 0ℎc0v 29273 Sℋ csh 29277 +ℋ cph 29280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-hilex 29348 ax-hfvadd 29349 ax-hvcom 29350 ax-hvass 29351 ax-hv0cl 29352 ax-hvaddid 29353 ax-hfvmul 29354 ax-hvmulid 29355 ax-hvdistr2 29358 ax-hvmul0 29359 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5486 df-po 5500 df-so 5501 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-er 8487 df-en 8723 df-dom 8724 df-sdom 8725 df-pnf 11000 df-mnf 11001 df-ltxr 11003 df-sub 11196 df-neg 11197 df-grpo 28842 df-ablo 28894 df-hvsub 29320 df-sh 29556 df-shs 29657 |
This theorem is referenced by: shsval2i 29736 shjshsi 29841 spanuni 29893 |
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