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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 4735 | . . 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 18760 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝐵) → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦))) |
9 | 1, 2, 4, 8 | syl3anc 1366 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦))) |
10 | oveq2 7157 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑥 + 𝑦) = (𝑥 + 𝑋)) | |
11 | 10 | eqeq2d 2831 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑍 = (𝑥 + 𝑦) ↔ 𝑍 = (𝑥 + 𝑋))) |
12 | 11 | rexsng 4607 | . . . 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 3296 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ {𝑋}𝑍 = (𝑥 + 𝑦) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋))) |
15 | 9, 14 | bitrd 281 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ⊆ wss 3929 {csn 4560 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 LSSumclsm 18754 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-lsm 18756 |
This theorem is referenced by: lsmsnorb 30967 lsmsnpridl 30971 mxidlprm 31001 |
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