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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 19074 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
8 | 5, 7 | eqeltrid 2836 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐺)) |
9 | qus0subg.e | . . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑆) | |
10 | 3, 9 | eqger 19101 | . . . . . 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 19092 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) |
14 | 5, 13 | eqeltrid 2836 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (NrmSGrp‘𝐺)) |
15 | eqid 2731 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 3, 9, 15 | eqgcpbl 19105 | . . . . . 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 18869 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(+g‘𝐺)𝑞) ∈ 𝐵) |
19 | 18 | 3expb 1119 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (𝑝(+g‘𝐺)𝑞) ∈ 𝐵) |
20 | eqid 2731 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
21 | 2, 4, 11, 12, 17, 19, 15, 20 | qusaddval 17506 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = [(𝑎(+g‘𝐺)𝑏)] ∼ ) |
22 | 21 | 3expb 1119 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = [(𝑎(+g‘𝐺)𝑏)] ∼ ) |
23 | 6, 5, 3, 9 | eqg0subgecsn 19119 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ = {𝑎}) |
24 | 23 | adantrr 714 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ = {𝑎}) |
25 | 6, 5, 3, 9 | eqg0subgecsn 19119 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ = {𝑏}) |
26 | 25 | adantrl 713 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ = {𝑏}) |
27 | 24, 26 | oveq12d 7430 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = ({𝑎} (+g‘𝑈){𝑏})) |
28 | 3, 15 | grpcl 18869 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
29 | 28 | 3expb 1119 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
30 | 6, 5, 3, 9 | eqg0subgecsn 19119 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) → [(𝑎(+g‘𝐺)𝑏)] ∼ = {(𝑎(+g‘𝐺)𝑏)}) |
31 | 29, 30 | syldan 590 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎(+g‘𝐺)𝑏)] ∼ = {(𝑎(+g‘𝐺)𝑏)}) |
32 | 22, 27, 31 | 3eqtr3d 2779 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)}) |
33 | 32 | ralrimivva 3199 | 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 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {csn 4628 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Er wer 8706 [cec 8707 Basecbs 17151 +gcplusg 17204 0gc0g 17392 /s cqus 17458 Grpcgrp 18861 SubGrpcsubg 19043 NrmSGrpcnsg 19044 ~QG cqg 19045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-ec 8711 df-qs 8715 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-0g 17394 df-imas 17461 df-qus 17462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-nsg 19047 df-eqg 19048 |
This theorem is referenced by: (None) |
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