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Mirrors > Home > MPE Home > Th. List > m2detleiblem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for m2detleib 22114. (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 19984 | . 2 ⊢ (Base‘𝑅) = (Base‘𝐺) |
4 | 1 | ringmgp 20052 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
5 | 4 | 3ad2ant1 1134 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd) |
6 | 2eluzge1 12873 | . . 3 ⊢ 2 ∈ (ℤ≥‘1) | |
7 | 6 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 2 ∈ (ℤ≥‘1)) |
8 | 1z 12587 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | fzpr 13551 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
11 | 1p1e2 12332 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 11 | preq2i 4739 | . . . . 5 ⊢ {1, (1 + 1)} = {1, 2} |
13 | 10, 12 | eqtri 2761 | . . . 4 ⊢ (1...(1 + 1)) = {1, 2} |
14 | df-2 12270 | . . . . 5 ⊢ 2 = (1 + 1) | |
15 | 14 | oveq2i 7414 | . . . 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 21945 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) |
23 | 3, 5, 7, 18, 22 | gsummptfzcl 19828 | 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 4628 ↦ cmpt 5229 ‘cfv 6539 (class class class)co 7403 1c1 11106 + caddc 11108 2c2 12262 ℤcz 12553 ℤ≥cuz 12817 ...cfz 13479 Basecbs 17139 Σg cgsu 17381 Mndcmnd 18620 SymGrpcsymg 19226 mulGrpcmgp 19978 Ringcrg 20046 Mat cmat 21888 |
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 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-fz 13480 df-seq 13962 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 df-ple 17212 df-ds 17214 df-hom 17216 df-cco 17217 df-0g 17382 df-gsum 17383 df-prds 17388 df-pws 17390 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-efmnd 18745 df-symg 19227 df-mgp 19979 df-ring 20048 df-sra 20772 df-rgmod 20773 df-dsmm 21270 df-frlm 21285 df-mat 21889 |
This theorem is referenced by: m2detleib 22114 |
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