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Mirrors > Home > HSE Home > Th. List > shless | Structured version Visualization version GIF version |
Description: Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shless | ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +ℋ 𝐶) ⊆ (𝐵 +ℋ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexv 4010 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧))) | |
2 | 1 | adantl 483 | . . 3 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧))) |
3 | simpl1 1192 | . . . 4 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ Sℋ ) | |
4 | simpl3 1194 | . . . 4 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐶 ∈ Sℋ ) | |
5 | shsel 30085 | . . . 4 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → (𝑥 ∈ (𝐴 +ℋ 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧))) | |
6 | 3, 4, 5 | syl2anc 585 | . . 3 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (𝐴 +ℋ 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧))) |
7 | simpl2 1193 | . . . 4 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ Sℋ ) | |
8 | shsel 30085 | . . . 4 ⊢ ((𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → (𝑥 ∈ (𝐵 +ℋ 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧))) | |
9 | 7, 4, 8 | syl2anc 585 | . . 3 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (𝐵 +ℋ 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = (𝑦 +ℎ 𝑧))) |
10 | 2, 6, 9 | 3imtr4d 294 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (𝐴 +ℋ 𝐶) → 𝑥 ∈ (𝐵 +ℋ 𝐶))) |
11 | 10 | ssrdv 3949 | 1 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +ℋ 𝐶) ⊆ (𝐵 +ℋ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 ⊆ wss 3909 (class class class)co 7352 +ℎ cva 29691 Sℋ csh 29699 +ℋ cph 29702 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-hilex 29770 ax-hfvadd 29771 ax-hvcom 29772 ax-hvass 29773 ax-hv0cl 29774 ax-hvaddid 29775 ax-hfvmul 29776 ax-hvmulid 29777 ax-hvdistr2 29780 ax-hvmul0 29781 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-po 5544 df-so 5545 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-ltxr 11153 df-sub 11346 df-neg 11347 df-grpo 29264 df-ablo 29316 df-hvsub 29742 df-sh 29978 df-shs 30079 |
This theorem is referenced by: shlessi 30148 pjpjpre 30190 chscllem1 30408 chscllem2 30409 chscllem3 30410 |
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