Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 18517. (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 2758 | . . 3 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmsnorb2.3 | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 18847 | . 2 ⊢ (𝐴(LSSum‘(oppg‘𝐺)){𝑋}) = ({𝑋} ⊕ 𝐴) |
4 | lsmsnorb2.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 1, 4 | oppgbas 18559 | . . 3 ⊢ 𝐵 = (Base‘(oppg‘𝐺)) |
6 | eqid 2758 | . . 3 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺)) | |
7 | eqid 2758 | . . 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 18557 | . . . . . . . 8 ⊢ (𝑔(+g‘(oppg‘𝐺))𝑥) = (𝑥 + 𝑔) |
11 | 10 | eqeq1i 2763 | . . . . . . 7 ⊢ ((𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦 ↔ (𝑥 + 𝑔) = 𝑦) |
12 | 11 | rexbii 3175 | . . . . . 6 ⊢ (∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦 ↔ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦) |
13 | 12 | anbi2i 625 | . . . . 5 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)) |
14 | 13 | opabbii 5103 | . . . 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 2784 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦)} |
16 | lsmsnorb2.5 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
17 | 1 | oppgmnd 18562 | . . . 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 31112 | . 2 ⊢ (𝜑 → (𝐴(LSSum‘(oppg‘𝐺)){𝑋}) = [𝑋] ∼ ) |
22 | 3, 21 | eqtr3id 2807 | 1 ⊢ (𝜑 → ({𝑋} ⊕ 𝐴) = [𝑋] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3071 ⊆ wss 3860 {csn 4525 {cpr 4527 {copab 5098 ‘cfv 6340 (class class class)co 7156 [cec 8303 Basecbs 16554 +gcplusg 16636 Mndcmnd 17990 oppgcoppg 18553 LSSumclsm 18839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-tpos 7908 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-er 8305 df-ec 8307 df-en 8541 df-dom 8542 df-sdom 8543 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-plusg 16649 df-0g 16786 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-oppg 18554 df-lsm 18841 |
This theorem is referenced by: quslsm 31126 |
Copyright terms: Public domain | W3C validator |