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Mirrors > Home > MPE Home > Th. List > m2detleiblem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for m2detleib 21395. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 1-Jan-2019.) |
Ref | Expression |
---|---|
m2detleiblem2.n | ⊢ 𝑁 = {1, 2} |
m2detleiblem2.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
m2detleiblem2.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
m2detleiblem2.b | ⊢ 𝐵 = (Base‘𝐴) |
m2detleiblem2.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
m2detleiblem2 | ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m2detleiblem2.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
2 | eqid 2739 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 19377 | . 2 ⊢ (Base‘𝑅) = (Base‘𝐺) |
4 | 1 | ringmgp 19435 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
5 | 4 | 3ad2ant1 1134 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd) |
6 | 2eluzge1 12389 | . . 3 ⊢ 2 ∈ (ℤ≥‘1) | |
7 | 6 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 2 ∈ (ℤ≥‘1)) |
8 | 1z 12106 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | fzpr 13066 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
11 | 1p1e2 11854 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 11 | preq2i 4638 | . . . . 5 ⊢ {1, (1 + 1)} = {1, 2} |
13 | 10, 12 | eqtri 2762 | . . . 4 ⊢ (1...(1 + 1)) = {1, 2} |
14 | df-2 11792 | . . . . 5 ⊢ 2 = (1 + 1) | |
15 | 14 | oveq2i 7194 | . . . 4 ⊢ (1...2) = (1...(1 + 1)) |
16 | m2detleiblem2.n | . . . 4 ⊢ 𝑁 = {1, 2} | |
17 | 13, 15, 16 | 3eqtr4ri 2773 | . . 3 ⊢ 𝑁 = (1...2) |
18 | 17 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 𝑁 = (1...2)) |
19 | m2detleiblem2.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
20 | m2detleiblem2.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
21 | m2detleiblem2.p | . . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
22 | 19, 20, 21 | matepmcl 21226 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) |
23 | 3, 5, 7, 18, 22 | gsummptfzcl 19221 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 {cpr 4528 ↦ cmpt 5120 ‘cfv 6350 (class class class)co 7183 1c1 10629 + caddc 10631 2c2 11784 ℤcz 12075 ℤ≥cuz 12337 ...cfz 12994 Basecbs 16599 Σg cgsu 16830 Mndcmnd 18040 SymGrpcsymg 18626 mulGrpcmgp 19371 Ringcrg 19429 Mat cmat 21171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-ot 4535 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-sup 8992 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-fz 12995 df-seq 13474 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-hom 16705 df-cco 16706 df-0g 16831 df-gsum 16832 df-prds 16837 df-pws 16839 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-efmnd 18163 df-symg 18627 df-mgp 19372 df-ring 19431 df-sra 20076 df-rgmod 20077 df-dsmm 20561 df-frlm 20576 df-mat 21172 |
This theorem is referenced by: m2detleib 21395 |
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