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Mirrors > Home > MPE Home > Th. List > Mathboxes > qustrivr | Structured version Visualization version GIF version |
Description: Converse of qustriv 31856. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
Ref | Expression |
---|---|
qustrivr.1 | ⊢ 𝐵 = (Base‘𝐺) |
qustrivr.2 | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐻)) |
Ref | Expression |
---|---|
qustrivr | ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → 𝐻 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qustrivr.2 | . . . . . . 7 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐻)) | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → 𝑄 = (𝐺 /s (𝐺 ~QG 𝐻))) |
3 | qustrivr.1 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → 𝐵 = (Base‘𝐺)) |
5 | ovexd 7373 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝐺 ~QG 𝐻) ∈ V) | |
6 | simpl 483 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp) | |
7 | 2, 4, 5, 6 | qusbas 17354 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝐵 / (𝐺 ~QG 𝐻)) = (Base‘𝑄)) |
8 | 7 | 3adant3 1131 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → (𝐵 / (𝐺 ~QG 𝐻)) = (Base‘𝑄)) |
9 | simp3 1137 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → (Base‘𝑄) = {𝐻}) | |
10 | 8, 9 | eqtrd 2776 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → (𝐵 / (𝐺 ~QG 𝐻)) = {𝐻}) |
11 | 10 | unieqd 4867 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → ∪ (𝐵 / (𝐺 ~QG 𝐻)) = ∪ {𝐻}) |
12 | eqid 2736 | . . . . . 6 ⊢ (𝐺 ~QG 𝐻) = (𝐺 ~QG 𝐻) | |
13 | 3, 12 | eqger 18903 | . . . . 5 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐻) Er 𝐵) |
14 | 13 | adantl 482 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → (𝐺 ~QG 𝐻) Er 𝐵) |
15 | 14, 5 | uniqs2 8640 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ∪ (𝐵 / (𝐺 ~QG 𝐻)) = 𝐵) |
16 | 15 | 3adant3 1131 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → ∪ (𝐵 / (𝐺 ~QG 𝐻)) = 𝐵) |
17 | unisng 4874 | . . 3 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∪ {𝐻} = 𝐻) | |
18 | 17 | 3ad2ant2 1133 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → ∪ {𝐻} = 𝐻) |
19 | 11, 16, 18 | 3eqtr3rd 2785 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → 𝐻 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3441 {csn 4574 ∪ cuni 4853 ‘cfv 6480 (class class class)co 7338 Er wer 8567 / cqs 8569 Basecbs 17010 /s cqus 17314 Grpcgrp 18674 SubGrpcsubg 18846 ~QG cqg 18848 |
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 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-om 7782 df-1st 7900 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-1o 8368 df-er 8570 df-ec 8572 df-qs 8576 df-en 8806 df-dom 8807 df-sdom 8808 df-fin 8809 df-sup 9300 df-inf 9301 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-nn 12076 df-2 12138 df-3 12139 df-4 12140 df-5 12141 df-6 12142 df-7 12143 df-8 12144 df-9 12145 df-n0 12336 df-z 12422 df-dec 12540 df-uz 12685 df-fz 13342 df-struct 16946 df-sets 16963 df-slot 16981 df-ndx 16993 df-base 17011 df-ress 17040 df-plusg 17073 df-mulr 17074 df-sca 17076 df-vsca 17077 df-ip 17078 df-tset 17079 df-ple 17080 df-ds 17082 df-0g 17250 df-imas 17317 df-qus 17318 df-mgm 18424 df-sgrp 18473 df-mnd 18484 df-grp 18677 df-minusg 18678 df-subg 18849 df-eqg 18851 |
This theorem is referenced by: qsidomlem2 31926 |
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