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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 6916 | . . . 4 ⊢ 𝐾 ∈ V |
3 | mat1rhmval.o | . . . . 5 ⊢ 𝑂 = ⟨𝐸, 𝐸⟩ | |
4 | opex 5470 | . . . . 5 ⊢ ⟨𝐸, 𝐸⟩ ∈ V | |
5 | 3, 4 | eqeltri 2825 | . . . 4 ⊢ 𝑂 ∈ V |
6 | 2, 5 | pm3.2i 469 | . . 3 ⊢ (𝐾 ∈ V ∧ 𝑂 ∈ V) |
7 | vex 3477 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 5, 7 | xpsn 7156 | . . . . . 6 ⊢ ({𝑂} × {𝑥}) = {⟨𝑂, 𝑥⟩} |
9 | 8 | eqcomi 2737 | . . . . 5 ⊢ {⟨𝑂, 𝑥⟩} = ({𝑂} × {𝑥}) |
10 | 9 | mpteq2i 5257 | . . . 4 ⊢ (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩}) = (𝑥 ∈ 𝐾 ↦ ({𝑂} × {𝑥})) |
11 | 10 | mapsnf1o 8964 | . . 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 2729 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐾 = 𝐾) | |
16 | mat1rhmval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
17 | 3 | sneqi 4643 | . . . . . . 7 ⊢ {𝑂} = {⟨𝐸, 𝐸⟩} |
18 | simpr 483 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
19 | xpsng 7154 | . . . . . . . 8 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩}) | |
20 | 18, 19 | sylancom 586 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩}) |
21 | 17, 20 | eqtr4id 2787 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → {𝑂} = ({𝐸} × {𝐸})) |
22 | 21 | oveq2d 7442 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m {𝑂}) = (𝐾 ↑m ({𝐸} × {𝐸}))) |
23 | snfi 9075 | . . . . . 6 ⊢ {𝐸} ∈ Fin | |
24 | simpl 481 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝑅 ∈ Ring) | |
25 | mat1rhmval.a | . . . . . . 7 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
26 | 25, 1 | matbas2 22343 | . . . . . 6 ⊢ (({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑m ({𝐸} × {𝐸})) = (Base‘𝐴)) |
27 | 23, 24, 26 | sylancr 585 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m ({𝐸} × {𝐸})) = (Base‘𝐴)) |
28 | 22, 27 | eqtrd 2768 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m {𝑂}) = (Base‘𝐴)) |
29 | 16, 28 | eqtr4id 2787 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐵 = (𝐾 ↑m {𝑂})) |
30 | 14, 15, 29 | f1oeq123d 6838 | . 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 256 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 ⟨cop 4638 ↦ cmpt 5235 × cxp 5680 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 ↑m cmap 8851 Fincfn 8970 Basecbs 17187 Ringcrg 20180 Mat cmat 22327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-pws 17438 df-sra 21065 df-rgmod 21066 df-dsmm 21673 df-frlm 21688 df-mat 22328 |
This theorem is referenced by: mat1f 22404 mat1rngiso 22408 |
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