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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmsnorb2 | Structured version Visualization version GIF version |
Description: The sumset of a single element with a group is the element's orbit by the group action. See gaorb 19321. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
Ref | Expression |
---|---|
lsmsnorb2.1 | ⊢ 𝐵 = (Base‘𝐺) |
lsmsnorb2.2 | ⊢ + = (+g‘𝐺) |
lsmsnorb2.3 | ⊢ ⊕ = (LSSum‘𝐺) |
lsmsnorb2.4 | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)} |
lsmsnorb2.5 | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
lsmsnorb2.6 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
lsmsnorb2.7 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
lsmsnorb2 | ⊢ (𝜑 → ({𝑋} ⊕ 𝐴) = [𝑋] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmsnorb2.3 | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 19656 | . 2 ⊢ (𝐴(LSSum‘(oppg‘𝐺)){𝑋}) = ({𝑋} ⊕ 𝐴) |
4 | lsmsnorb2.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 1, 4 | oppgbas 19366 | . . 3 ⊢ 𝐵 = (Base‘(oppg‘𝐺)) |
6 | eqid 2736 | . . 3 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺)) | |
7 | eqid 2736 | . . 3 ⊢ (LSSum‘(oppg‘𝐺)) = (LSSum‘(oppg‘𝐺)) | |
8 | lsmsnorb2.4 | . . . 4 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)} | |
9 | lsmsnorb2.2 | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
10 | 9, 1, 6 | oppgplus 19363 | . . . . . . . 8 ⊢ (𝑔(+g‘(oppg‘𝐺))𝑥) = (𝑥 + 𝑔) |
11 | 10 | eqeq1i 2741 | . . . . . . 7 ⊢ ((𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦 ↔ (𝑥 + 𝑔) = 𝑦) |
12 | 11 | rexbii 3093 | . . . . . 6 ⊢ (∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦 ↔ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦) |
13 | 12 | anbi2i 623 | . . . . 5 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)) |
14 | 13 | opabbii 5208 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦)} = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)} |
15 | 8, 14 | eqtr4i 2767 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦)} |
16 | lsmsnorb2.5 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
17 | 1 | oppgmnd 19369 | . . . 4 ⊢ (𝐺 ∈ Mnd → (oppg‘𝐺) ∈ Mnd) |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → (oppg‘𝐺) ∈ Mnd) |
19 | lsmsnorb2.6 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
20 | lsmsnorb2.7 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
21 | 5, 6, 7, 15, 18, 19, 20 | lsmsnorb 33406 | . 2 ⊢ (𝜑 → (𝐴(LSSum‘(oppg‘𝐺)){𝑋}) = [𝑋] ∼ ) |
22 | 3, 21 | eqtr3id 2790 | 1 ⊢ (𝜑 → ({𝑋} ⊕ 𝐴) = [𝑋] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3069 ⊆ wss 3950 {csn 4624 {cpr 4626 {copab 5203 ‘cfv 6559 (class class class)co 7429 [cec 8739 Basecbs 17243 +gcplusg 17293 Mndcmnd 18743 oppgcoppg 19359 LSSumclsm 19648 |
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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-ec 8743 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-plusg 17306 df-0g 17482 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-oppg 19360 df-lsm 19650 |
This theorem is referenced by: quslsm 33420 |
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