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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 28994 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
3 | ax-hvaddid 28784 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ 0ℎ) = 𝑥) | |
4 | 3 | eqcomd 2830 | . . . . . 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 28996 | . . . . . . 7 ⊢ (𝐵 ∈ Sℋ → 0ℎ ∈ 𝐵) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐵 |
9 | rspceov 7206 | . . . . . 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 28994 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ) |
13 | hvaddid2 28803 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
14 | 13 | eqcomd 2830 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → 𝑥 = (0ℎ +ℎ 𝑥)) |
15 | 12, 14 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = (0ℎ +ℎ 𝑥)) |
16 | sh0 28996 | . . . . . . 7 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
17 | 1, 16 | ax-mp 5 | . . . . . 6 ⊢ 0ℎ ∈ 𝐴 |
18 | rspceov 7206 | . . . . . 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 4128 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
23 | 1, 6 | shseli 29096 | . . 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 3974 | 1 ⊢ (𝐴 ∪ 𝐵) ⊆ (𝐴 +ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ∪ cun 3937 ⊆ wss 3939 (class class class)co 7159 ℋchba 28699 +ℎ cva 28700 0ℎc0v 28704 Sℋ csh 28708 +ℋ cph 28711 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvdistr2 28789 ax-hvmul0 28790 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 df-grpo 28273 df-ablo 28325 df-hvsub 28751 df-sh 28987 df-shs 29088 |
This theorem is referenced by: shsval2i 29167 shjshsi 29272 spanuni 29324 |
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