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Mirrors > Home > MPE Home > Th. List > m2detleiblem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for m2detleib 22133. (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 2733 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 19993 | . 2 ⊢ (Base‘𝑅) = (Base‘𝐺) |
4 | 1 | ringmgp 20062 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
5 | 4 | 3ad2ant1 1134 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd) |
6 | 2eluzge1 12878 | . . 3 ⊢ 2 ∈ (ℤ≥‘1) | |
7 | 6 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 2 ∈ (ℤ≥‘1)) |
8 | 1z 12592 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | fzpr 13556 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
11 | 1p1e2 12337 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 11 | preq2i 4742 | . . . . 5 ⊢ {1, (1 + 1)} = {1, 2} |
13 | 10, 12 | eqtri 2761 | . . . 4 ⊢ (1...(1 + 1)) = {1, 2} |
14 | df-2 12275 | . . . . 5 ⊢ 2 = (1 + 1) | |
15 | 14 | oveq2i 7420 | . . . 4 ⊢ (1...2) = (1...(1 + 1)) |
16 | m2detleiblem2.n | . . . 4 ⊢ 𝑁 = {1, 2} | |
17 | 13, 15, 16 | 3eqtr4ri 2772 | . . 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 21964 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) |
23 | 3, 5, 7, 18, 22 | gsummptfzcl 19837 | 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 2107 {cpr 4631 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 1c1 11111 + caddc 11113 2c2 12267 ℤcz 12558 ℤ≥cuz 12822 ...cfz 13484 Basecbs 17144 Σg cgsu 17386 Mndcmnd 18625 SymGrpcsymg 19234 mulGrpcmgp 19987 Ringcrg 20056 Mat cmat 21907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-seq 13967 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-efmnd 18750 df-symg 19235 df-mgp 19988 df-ring 20058 df-sra 20785 df-rgmod 20786 df-dsmm 21287 df-frlm 21302 df-mat 21908 |
This theorem is referenced by: m2detleib 22133 |
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