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Mirrors > Home > MPE Home > Th. List > xpsgrp | Structured version Visualization version GIF version |
Description: The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpsgrp.t | ⊢ 𝑇 = (𝑅 ×s 𝑆) |
Ref | Expression |
---|---|
xpsgrp | ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsgrp.t | . . 3 ⊢ 𝑇 = (𝑅 ×s 𝑆) | |
2 | eqid 2772 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2772 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | simpl 475 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑅 ∈ Grp) | |
5 | simpr 477 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑆 ∈ Grp) | |
6 | eqid 2772 | . . 3 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) | |
7 | eqid 2772 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2772 | . . 3 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 16691 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 = (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))) |
10 | 6 | xpsff1o2 16690 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) |
11 | 1, 2, 3, 4, 5, 6, 7, 8 | xpslem 16692 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))) |
12 | 11 | f1oeq3d 6435 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ((𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→ran (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) ↔ (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))) |
13 | 10, 12 | mpbii 225 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))) |
14 | f1ocnv 6450 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):((Base‘𝑅) × (Base‘𝑆))–1-1-onto→(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))–1-1-onto→((Base‘𝑅) × (Base‘𝑆))) | |
15 | f1of1 6437 | . . . 4 ⊢ (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))–1-1-onto→((Base‘𝑅) × (Base‘𝑆)) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))–1-1→((Base‘𝑅) × (Base‘𝑆))) | |
16 | 13, 14, 15 | 3syl 18 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))–1-1→((Base‘𝑅) × (Base‘𝑆))) |
17 | 2on 7906 | . . . . 5 ⊢ 2o ∈ On | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 2o ∈ On) |
19 | fvexd 6508 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → (Scalar‘𝑅) ∈ V) | |
20 | xpscf 16685 | . . . . 5 ⊢ (◡({𝑅} +𝑐 {𝑆}):2o⟶Grp ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp)) | |
21 | 20 | biimpri 220 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ◡({𝑅} +𝑐 {𝑆}):2o⟶Grp) |
22 | 8, 18, 19, 21 | prdsgrpd 17986 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ Grp) |
23 | eqid 2772 | . . . 4 ⊢ (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) | |
24 | eqid 2772 | . . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) | |
25 | 23, 24 | imasgrpf1 17993 | . . 3 ⊢ ((◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})):(Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))–1-1→((Base‘𝑅) × (Base‘𝑆)) ∧ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ Grp) → (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ∈ Grp) |
26 | 16, 22, 25 | syl2anc 576 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → (◡(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑆) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) ∈ Grp) |
27 | 9, 26 | eqeltrd 2860 | 1 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 Vcvv 3409 {csn 4435 × cxp 5398 ◡ccnv 5399 ran crn 5401 Oncon0 6023 ⟶wf 6178 –1-1→wf1 6179 –1-1-onto→wf1o 6181 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 2oc2o 7891 +𝑐 ccda 9379 Basecbs 16329 Scalarcsca 16414 Xscprds 16565 “s cimas 16623 ×s cxps 16625 Grpcgrp 17881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-hom 16435 df-cco 16436 df-0g 16561 df-prds 16567 df-imas 16627 df-xps 16629 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-minusg 17885 |
This theorem is referenced by: (None) |
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