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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 4809 | . . 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 19658 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦))) | 
| 9 | 1, 2, 4, 8 | syl3anc 1373 | . 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 2748 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋))) | 
| 12 | 11 | rexsng 4676 | . . . 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 3179 | . 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 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 LSSumclsm 19652 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-lsm 19654 | 
| This theorem is referenced by: lsmsnorb 33419 lsmsnpridl 33426 mxidlprm 33498 | 
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