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Mirrors > Home > MPE Home > Th. List > mat1f1o | Structured version Visualization version GIF version |
Description: There is a 1-1 function from a ring onto the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
Ref | Expression |
---|---|
mat1rhmval.k | ⊢ 𝐾 = (Base‘𝑅) |
mat1rhmval.a | ⊢ 𝐴 = ({𝐸} Mat 𝑅) |
mat1rhmval.b | ⊢ 𝐵 = (Base‘𝐴) |
mat1rhmval.o | ⊢ 𝑂 = 〈𝐸, 𝐸〉 |
mat1rhmval.f | ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) |
Ref | Expression |
---|---|
mat1f1o | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mat1rhmval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
2 | 1 | fvexi 6736 | . . . 4 ⊢ 𝐾 ∈ V |
3 | mat1rhmval.o | . . . . 5 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
4 | opex 5353 | . . . . 5 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
5 | 3, 4 | eqeltri 2834 | . . . 4 ⊢ 𝑂 ∈ V |
6 | 2, 5 | pm3.2i 474 | . . 3 ⊢ (𝐾 ∈ V ∧ 𝑂 ∈ V) |
7 | vex 3417 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 5, 7 | xpsn 6961 | . . . . . 6 ⊢ ({𝑂} × {𝑥}) = {〈𝑂, 𝑥〉} |
9 | 8 | eqcomi 2746 | . . . . 5 ⊢ {〈𝑂, 𝑥〉} = ({𝑂} × {𝑥}) |
10 | 9 | mpteq2i 5152 | . . . 4 ⊢ (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) = (𝑥 ∈ 𝐾 ↦ ({𝑂} × {𝑥})) |
11 | 10 | mapsnf1o 8625 | . . 3 ⊢ ((𝐾 ∈ V ∧ 𝑂 ∈ V) → (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}):𝐾–1-1-onto→(𝐾 ↑m {𝑂})) |
12 | 6, 11 | mp1i 13 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}):𝐾–1-1-onto→(𝐾 ↑m {𝑂})) |
13 | mat1rhmval.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) | |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉})) |
15 | eqidd 2738 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐾 = 𝐾) | |
16 | mat1rhmval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
17 | 3 | sneqi 4557 | . . . . . . 7 ⊢ {𝑂} = {〈𝐸, 𝐸〉} |
18 | simpr 488 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
19 | xpsng 6959 | . . . . . . . 8 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
20 | 18, 19 | sylancom 591 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) |
21 | 17, 20 | eqtr4id 2797 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → {𝑂} = ({𝐸} × {𝐸})) |
22 | 21 | oveq2d 7234 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m {𝑂}) = (𝐾 ↑m ({𝐸} × {𝐸}))) |
23 | snfi 8726 | . . . . . 6 ⊢ {𝐸} ∈ Fin | |
24 | simpl 486 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝑅 ∈ Ring) | |
25 | mat1rhmval.a | . . . . . . 7 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
26 | 25, 1 | matbas2 21323 | . . . . . 6 ⊢ (({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑m ({𝐸} × {𝐸})) = (Base‘𝐴)) |
27 | 23, 24, 26 | sylancr 590 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m ({𝐸} × {𝐸})) = (Base‘𝐴)) |
28 | 22, 27 | eqtrd 2777 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m {𝑂}) = (Base‘𝐴)) |
29 | 16, 28 | eqtr4id 2797 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐵 = (𝐾 ↑m {𝑂})) |
30 | 14, 15, 29 | f1oeq123d 6660 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐹:𝐾–1-1-onto→𝐵 ↔ (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}):𝐾–1-1-onto→(𝐾 ↑m {𝑂}))) |
31 | 12, 30 | mpbird 260 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3413 {csn 4546 〈cop 4552 ↦ cmpt 5140 × cxp 5554 –1-1-onto→wf1o 6384 ‘cfv 6385 (class class class)co 7218 ↑m cmap 8513 Fincfn 8631 Basecbs 16765 Ringcrg 19567 Mat cmat 21309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-ot 4555 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-om 7650 df-1st 7766 df-2nd 7767 df-supp 7909 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-er 8396 df-map 8515 df-ixp 8584 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-fsupp 8991 df-sup 9063 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-7 11903 df-8 11904 df-9 11905 df-n0 12096 df-z 12182 df-dec 12299 df-uz 12444 df-fz 13101 df-struct 16705 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-sca 16823 df-vsca 16824 df-ip 16825 df-tset 16826 df-ple 16827 df-ds 16829 df-hom 16831 df-cco 16832 df-0g 16951 df-prds 16957 df-pws 16959 df-sra 20214 df-rgmod 20215 df-dsmm 20699 df-frlm 20714 df-mat 21310 |
This theorem is referenced by: mat1f 21384 mat1rngiso 21388 |
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