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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 19182 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) |
8 | 5, 7 | eqeltrid 2843 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐺)) |
9 | qus0subg.e | . . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑆) | |
10 | 3, 9 | eqger 19209 | . . . . . 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 19200 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺)) |
14 | 5, 13 | eqeltrid 2843 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (NrmSGrp‘𝐺)) |
15 | eqid 2735 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 3, 9, 15 | eqgcpbl 19213 | . . . . . 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 18972 | . . . . . 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 2735 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
21 | 2, 4, 11, 12, 17, 19, 15, 20 | qusaddval 17600 | . . . 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 19228 | . . . . 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 19228 | . . . . 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 18972 | . . . . 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 19228 | . . . 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 2783 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)}) |
33 | 32 | ralrimivva 3200 | 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 1537 ∈ wcel 2106 ∀wral 3059 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Er wer 8741 [cec 8742 Basecbs 17245 +gcplusg 17298 0gc0g 17486 /s cqus 17552 Grpcgrp 18964 SubGrpcsubg 19151 NrmSGrpcnsg 19152 ~QG cqg 19153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 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 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 df-eqg 19156 |
This theorem is referenced by: (None) |
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