Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qusxpid | Structured version Visualization version GIF version |
Description: The Group quotient equivalence relation for the whole group is the cartesian product, i.e. all elements are in the same equivalence class. (Contributed by Thierry Arnoux, 16-Jan-2024.) |
Ref | Expression |
---|---|
qustriv.1 | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
qusxpid | ⊢ (𝐺 ∈ Grp → (𝐺 ~QG 𝐵) = (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qustriv.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | 1 | subgid 18545 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺)) |
3 | eqid 2737 | . . . 4 ⊢ (𝐺 ~QG 𝐵) = (𝐺 ~QG 𝐵) | |
4 | 1, 3 | eqger 18594 | . . 3 ⊢ (𝐵 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐵) Er 𝐵) |
5 | errel 8400 | . . 3 ⊢ ((𝐺 ~QG 𝐵) Er 𝐵 → Rel (𝐺 ~QG 𝐵)) | |
6 | 2, 4, 5 | 3syl 18 | . 2 ⊢ (𝐺 ∈ Grp → Rel (𝐺 ~QG 𝐵)) |
7 | relxp 5569 | . . 3 ⊢ Rel (𝐵 × 𝐵) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → Rel (𝐵 × 𝐵)) |
9 | df-3an 1091 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵)) | |
10 | simpl 486 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp) | |
11 | eqid 2737 | . . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
12 | 1, 11 | grpinvcl 18415 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵) |
13 | 12 | adantrr 717 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑥) ∈ 𝐵) |
14 | simprr 773 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) | |
15 | eqid 2737 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 1, 15 | grpcl 18373 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵) |
17 | 10, 13, 14, 16 | syl3anc 1373 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵) |
18 | 17 | ex 416 | . . . . 5 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵)) |
19 | 18 | pm4.71d 565 | . . . 4 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵))) |
20 | 9, 19 | bitr4id 293 | . . 3 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) |
21 | ssid 3923 | . . . 4 ⊢ 𝐵 ⊆ 𝐵 | |
22 | 1, 11, 15, 3 | eqgval 18593 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ⊆ 𝐵) → (𝑥(𝐺 ~QG 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵))) |
23 | 21, 22 | mpan2 691 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑥(𝐺 ~QG 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (((invg‘𝐺)‘𝑥)(+g‘𝐺)𝑦) ∈ 𝐵))) |
24 | brxp 5598 | . . . 4 ⊢ (𝑥(𝐵 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) | |
25 | 24 | a1i 11 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑥(𝐵 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) |
26 | 20, 23, 25 | 3bitr4d 314 | . 2 ⊢ (𝐺 ∈ Grp → (𝑥(𝐺 ~QG 𝐵)𝑦 ↔ 𝑥(𝐵 × 𝐵)𝑦)) |
27 | 6, 8, 26 | eqbrrdv 5663 | 1 ⊢ (𝐺 ∈ Grp → (𝐺 ~QG 𝐵) = (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 class class class wbr 5053 × cxp 5549 Rel wrel 5556 ‘cfv 6380 (class class class)co 7213 Er wer 8388 Basecbs 16760 +gcplusg 16802 Grpcgrp 18365 invgcminusg 18366 SubGrpcsubg 18537 ~QG cqg 18539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-minusg 18369 df-subg 18540 df-eqg 18542 |
This theorem is referenced by: qustriv 31274 |
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