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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 4770 | . . 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 19427 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦))) |
9 | 1, 2, 4, 8 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦))) |
10 | oveq2 7366 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋)) | |
11 | 10 | eqeq2d 2744 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋))) |
12 | 11 | rexsng 4636 | . . . 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 3172 | . 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 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 ⊆ wss 3911 {csn 4587 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 LSSumclsm 19421 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-lsm 19423 |
This theorem is referenced by: lsmsnorb 32220 lsmsnpridl 32227 mxidlprm 32285 |
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