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Mirrors > Home > MPE Home > Th. List > m2detleiblem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for m2detleib 22652. (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 2734 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 20157 | . 2 ⊢ (Base‘𝑅) = (Base‘𝐺) |
4 | 1 | ringmgp 20256 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
5 | 4 | 3ad2ant1 1132 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd) |
6 | 2eluzge1 12933 | . . 3 ⊢ 2 ∈ (ℤ≥‘1) | |
7 | 6 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 2 ∈ (ℤ≥‘1)) |
8 | 1z 12644 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | fzpr 13615 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
11 | 1p1e2 12388 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 11 | preq2i 4741 | . . . . 5 ⊢ {1, (1 + 1)} = {1, 2} |
13 | 10, 12 | eqtri 2762 | . . . 4 ⊢ (1...(1 + 1)) = {1, 2} |
14 | df-2 12326 | . . . . 5 ⊢ 2 = (1 + 1) | |
15 | 14 | oveq2i 7441 | . . . 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 22483 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) |
23 | 3, 5, 7, 18, 22 | gsummptfzcl 20001 | 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 1086 = wceq 1536 ∈ wcel 2105 {cpr 4632 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 1c1 11153 + caddc 11155 2c2 12318 ℤcz 12610 ℤ≥cuz 12875 ...cfz 13543 Basecbs 17244 Σg cgsu 17486 Mndcmnd 18759 SymGrpcsymg 19400 mulGrpcmgp 20151 Ringcrg 20250 Mat cmat 22426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-seq 14039 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-efmnd 18894 df-symg 19401 df-mgp 20152 df-ring 20252 df-sra 21189 df-rgmod 21190 df-dsmm 21769 df-frlm 21784 df-mat 22427 |
This theorem is referenced by: m2detleib 22652 |
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