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Mirrors > Home > MPE Home > Th. List > m2detleiblem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for m2detleib 21240. (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 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 19245 | . 2 ⊢ (Base‘𝑅) = (Base‘𝐺) |
4 | 1 | ringmgp 19303 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
5 | 4 | 3ad2ant1 1129 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ Mnd) |
6 | 2eluzge1 12295 | . . 3 ⊢ 2 ∈ (ℤ≥‘1) | |
7 | 6 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → 2 ∈ (ℤ≥‘1)) |
8 | 1z 12013 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | fzpr 12963 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...(1 + 1)) = {1, (1 + 1)}) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (1...(1 + 1)) = {1, (1 + 1)} |
11 | 1p1e2 11763 | . . . . . 6 ⊢ (1 + 1) = 2 | |
12 | 11 | preq2i 4673 | . . . . 5 ⊢ {1, (1 + 1)} = {1, 2} |
13 | 10, 12 | eqtri 2844 | . . . 4 ⊢ (1...(1 + 1)) = {1, 2} |
14 | df-2 11701 | . . . . 5 ⊢ 2 = (1 + 1) | |
15 | 14 | oveq2i 7167 | . . . 4 ⊢ (1...2) = (1...(1 + 1)) |
16 | m2detleiblem2.n | . . . 4 ⊢ 𝑁 = {1, 2} | |
17 | 13, 15, 16 | 3eqtr4ri 2855 | . . 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 21071 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) |
23 | 3, 5, 7, 18, 22 | gsummptfzcl 19089 | 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 1083 = wceq 1537 ∈ wcel 2114 {cpr 4569 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 1c1 10538 + caddc 10540 2c2 11693 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 Basecbs 16483 Σg cgsu 16714 Mndcmnd 17911 SymGrpcsymg 18495 mulGrpcmgp 19239 Ringcrg 19297 Mat cmat 21016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-seq 13371 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-efmnd 18034 df-symg 18496 df-mgp 19240 df-ring 19299 df-sra 19944 df-rgmod 19945 df-dsmm 20876 df-frlm 20891 df-mat 21017 |
This theorem is referenced by: m2detleib 21240 |
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