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Mirrors > Home > MPE Home > Th. List > qus0subgadd | Structured version Visualization version GIF version |
Description: The addition in a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025.) |
Ref | Expression |
---|---|
qus0subg.0 | ⊢ 0 = (0g‘𝐺) |
qus0subg.s | ⊢ 𝑆 = { 0 } |
qus0subg.e | ⊢ ∼ = (𝐺 ~QG 𝑆) |
qus0subg.u | ⊢ 𝑈 = (𝐺 /s ∼ ) |
qus0subg.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
qus0subgadd | ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qus0subg.u | . . . . . 6 ⊢ 𝑈 = (𝐺 /s ∼ ) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝑈 = (𝐺 /s ∼ )) |
3 | qus0subg.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝐵 = (Base‘𝐺)) |
5 | qus0subg.s | . . . . . . 7 ⊢ 𝑆 = { 0 } | |
6 | qus0subg.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝐺) | |
7 | 6 | 0subg 19105 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
8 | 5, 7 | eqeltrid 2829 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐺)) |
9 | qus0subg.e | . . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑆) | |
10 | 3, 9 | eqger 19132 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∼ Er 𝐵) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ Grp → ∼ Er 𝐵) |
12 | id 22 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
13 | 6 | 0nsg 19123 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) |
14 | 5, 13 | eqeltrid 2829 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (NrmSGrp‘𝐺)) |
15 | eqid 2725 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 3, 9, 15 | eqgcpbl 19136 | . . . . . 6 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ((𝑥 ∼ 𝑝 ∧ 𝑦 ∼ 𝑞) → (𝑥(+g‘𝐺)𝑦) ∼ (𝑝(+g‘𝐺)𝑞))) |
17 | 14, 16 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ Grp → ((𝑥 ∼ 𝑝 ∧ 𝑦 ∼ 𝑞) → (𝑥(+g‘𝐺)𝑦) ∼ (𝑝(+g‘𝐺)𝑞))) |
18 | 3, 15 | grpcl 18897 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(+g‘𝐺)𝑞) ∈ 𝐵) |
19 | 18 | 3expb 1117 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (𝑝(+g‘𝐺)𝑞) ∈ 𝐵) |
20 | eqid 2725 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
21 | 2, 4, 11, 12, 17, 19, 15, 20 | qusaddval 17529 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = [(𝑎(+g‘𝐺)𝑏)] ∼ ) |
22 | 21 | 3expb 1117 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = [(𝑎(+g‘𝐺)𝑏)] ∼ ) |
23 | 6, 5, 3, 9 | eqg0subgecsn 19151 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ = {𝑎}) |
24 | 23 | adantrr 715 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ = {𝑎}) |
25 | 6, 5, 3, 9 | eqg0subgecsn 19151 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ = {𝑏}) |
26 | 25 | adantrl 714 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ = {𝑏}) |
27 | 24, 26 | oveq12d 7431 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = ({𝑎} (+g‘𝑈){𝑏})) |
28 | 3, 15 | grpcl 18897 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
29 | 28 | 3expb 1117 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
30 | 6, 5, 3, 9 | eqg0subgecsn 19151 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) → [(𝑎(+g‘𝐺)𝑏)] ∼ = {(𝑎(+g‘𝐺)𝑏)}) |
31 | 29, 30 | syldan 589 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎(+g‘𝐺)𝑏)] ∼ = {(𝑎(+g‘𝐺)𝑏)}) |
32 | 22, 27, 31 | 3eqtr3d 2773 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)}) |
33 | 32 | ralrimivva 3191 | 1 ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 {csn 4625 class class class wbr 5144 ‘cfv 6543 (class class class)co 7413 Er wer 8715 [cec 8716 Basecbs 17174 +gcplusg 17227 0gc0g 17415 /s cqus 17481 Grpcgrp 18889 SubGrpcsubg 19074 NrmSGrpcnsg 19075 ~QG cqg 19076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-0g 17417 df-imas 17484 df-qus 17485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-nsg 19078 df-eqg 19079 |
This theorem is referenced by: (None) |
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