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Mirrors > Home > MPE Home > Th. List > lsmless2x | Structured version Visualization version GIF version |
Description: Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsmless2.v | ⊢ 𝐵 = (Base‘𝐺) |
lsmless2.s | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmless2x | ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexv 3955 | . . . . 5 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧))) | |
2 | 1 | reximdv 3236 | . . . 4 ⊢ (𝑇 ⊆ 𝑈 → (∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
3 | 2 | adantl 482 | . . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧) → ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
4 | simpl1 1184 | . . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝐺 ∈ 𝑉) | |
5 | simpl2 1185 | . . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑅 ⊆ 𝐵) | |
6 | simpr 485 | . . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈) | |
7 | simpl3 1186 | . . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑈 ⊆ 𝐵) | |
8 | 6, 7 | sstrd 3899 | . . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝐵) |
9 | lsmless2.v | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
10 | eqid 2795 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | lsmless2.s | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
12 | 9, 10, 11 | lsmelvalx 18495 | . . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑥 ∈ (𝑅 ⊕ 𝑇) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
13 | 4, 5, 8, 12 | syl3anc 1364 | . . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑇) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑇 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
14 | 9, 10, 11 | lsmelvalx 18495 | . . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑥 ∈ (𝑅 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
15 | 14 | adantr 481 | . . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑈 𝑥 = (𝑦(+g‘𝐺)𝑧))) |
16 | 3, 13, 15 | 3imtr4d 295 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑥 ∈ (𝑅 ⊕ 𝑇) → 𝑥 ∈ (𝑅 ⊕ 𝑈))) |
17 | 16 | ssrdv 3895 | 1 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 ⊆ wss 3859 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 +gcplusg 16394 LSSumclsm 18489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-lsm 18491 |
This theorem is referenced by: lsmless2 18515 lsmssspx 19550 |
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