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Mirrors > Home > MPE Home > Th. List > Mathboxes > elgrplsmsn | Structured version Visualization version GIF version |
Description: Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
Ref | Expression |
---|---|
elgrplsmsn.1 | ⊢ 𝐵 = (Base‘𝐺) |
elgrplsmsn.2 | ⊢ + = (+g‘𝐺) |
elgrplsmsn.3 | ⊢ ⊕ = (LSSum‘𝐺) |
elgrplsmsn.4 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
elgrplsmsn.5 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
elgrplsmsn.6 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
elgrplsmsn | ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elgrplsmsn.4 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
2 | elgrplsmsn.5 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
3 | elgrplsmsn.6 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
4 | 3 | snssd 4814 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝐵) |
5 | elgrplsmsn.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
6 | elgrplsmsn.2 | . . . 4 ⊢ + = (+g‘𝐺) | |
7 | elgrplsmsn.3 | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
8 | 5, 6, 7 | lsmelvalx 19673 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦))) |
9 | 1, 2, 4, 8 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦))) |
10 | oveq2 7439 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋)) | |
11 | 10 | eqeq2d 2746 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋))) |
12 | 11 | rexsng 4681 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋))) |
13 | 3, 12 | syl 17 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋))) |
14 | 13 | rexbidv 3177 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋))) |
15 | 9, 14 | bitrd 279 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 LSSumclsm 19667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-lsm 19669 |
This theorem is referenced by: lsmsnorb 33399 lsmsnpridl 33406 mxidlprm 33478 |
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