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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 19323. (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 2733 | . . 3 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
2 | lsmsnorb2.3 | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
3 | 1, 2 | oppglsm 19660 | . 2 ⊢ (𝐴(LSSum‘(oppg‘𝐺)){𝑋}) = ({𝑋} ⊕ 𝐴) |
4 | lsmsnorb2.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 1, 4 | oppgbas 19368 | . . 3 ⊢ 𝐵 = (Base‘(oppg‘𝐺)) |
6 | eqid 2733 | . . 3 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺)) | |
7 | eqid 2733 | . . 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 19365 | . . . . . . . 8 ⊢ (𝑔(+g‘(oppg‘𝐺))𝑥) = (𝑥 + 𝑔) |
11 | 10 | eqeq1i 2738 | . . . . . . 7 ⊢ ((𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦 ↔ (𝑥 + 𝑔) = 𝑦) |
12 | 11 | rexbii 3090 | . . . . . 6 ⊢ (∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦 ↔ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦) |
13 | 12 | anbi2i 622 | . . . . 5 ⊢ (({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)) |
14 | 13 | opabbii 5216 | . . . 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 2764 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔(+g‘(oppg‘𝐺))𝑥) = 𝑦)} |
16 | lsmsnorb2.5 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
17 | 1 | oppgmnd 19373 | . . . 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 33362 | . 2 ⊢ (𝜑 → (𝐴(LSSum‘(oppg‘𝐺)){𝑋}) = [𝑋] ∼ ) |
22 | 3, 21 | eqtr3id 2787 | 1 ⊢ (𝜑 → ({𝑋} ⊕ 𝐴) = [𝑋] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ∃wrex 3066 ⊆ wss 3963 {csn 4630 {cpr 4632 {copab 5211 ‘cfv 6558 (class class class)co 7425 [cec 8736 Basecbs 17234 +gcplusg 17287 Mndcmnd 18748 oppgcoppg 19361 LSSumclsm 19652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-tpos 8244 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-er 8738 df-ec 8740 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-nn 12258 df-2 12320 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-plusg 17300 df-0g 17477 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-oppg 19362 df-lsm 19654 |
This theorem is referenced by: quslsm 33376 |
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