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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 19169 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) | 
| 8 | 5, 7 | eqeltrid 2845 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐺)) | 
| 9 | qus0subg.e | . . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑆) | |
| 10 | 3, 9 | eqger 19196 | . . . . . 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 19187 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) | 
| 14 | 5, 13 | eqeltrid 2845 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (NrmSGrp‘𝐺)) | 
| 15 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 16 | 3, 9, 15 | eqgcpbl 19200 | . . . . . 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 18959 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(+g‘𝐺)𝑞) ∈ 𝐵) | 
| 19 | 18 | 3expb 1121 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (𝑝(+g‘𝐺)𝑞) ∈ 𝐵) | 
| 20 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
| 21 | 2, 4, 11, 12, 17, 19, 15, 20 | qusaddval 17598 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = [(𝑎(+g‘𝐺)𝑏)] ∼ ) | 
| 22 | 21 | 3expb 1121 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = [(𝑎(+g‘𝐺)𝑏)] ∼ ) | 
| 23 | 6, 5, 3, 9 | eqg0subgecsn 19215 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ = {𝑎}) | 
| 24 | 23 | adantrr 717 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ = {𝑎}) | 
| 25 | 6, 5, 3, 9 | eqg0subgecsn 19215 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ = {𝑏}) | 
| 26 | 25 | adantrl 716 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ = {𝑏}) | 
| 27 | 24, 26 | oveq12d 7449 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑈)[𝑏] ∼ ) = ({𝑎} (+g‘𝑈){𝑏})) | 
| 28 | 3, 15 | grpcl 18959 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) | 
| 29 | 28 | 3expb 1121 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) | 
| 30 | 6, 5, 3, 9 | eqg0subgecsn 19215 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) → [(𝑎(+g‘𝐺)𝑏)] ∼ = {(𝑎(+g‘𝐺)𝑏)}) | 
| 31 | 29, 30 | syldan 591 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎(+g‘𝐺)𝑏)] ∼ = {(𝑎(+g‘𝐺)𝑏)}) | 
| 32 | 22, 27, 31 | 3eqtr3d 2785 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)}) | 
| 33 | 32 | ralrimivva 3202 | 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 2108 ∀wral 3061 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Er wer 8742 [cec 8743 Basecbs 17247 +gcplusg 17297 0gc0g 17484 /s cqus 17550 Grpcgrp 18951 SubGrpcsubg 19138 NrmSGrpcnsg 19139 ~QG cqg 19140 | 
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 | 
| This theorem is referenced by: (None) | 
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