Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6679 | . . . 4 ⊢ 𝐾 ∈ V |
3 | mat1rhmval.o | . . . . 5 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
4 | opex 5349 | . . . . 5 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
5 | 3, 4 | eqeltri 2909 | . . . 4 ⊢ 𝑂 ∈ V |
6 | 2, 5 | pm3.2i 473 | . . 3 ⊢ (𝐾 ∈ V ∧ 𝑂 ∈ V) |
7 | vex 3498 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 5, 7 | xpsn 6898 | . . . . . 6 ⊢ ({𝑂} × {𝑥}) = {〈𝑂, 𝑥〉} |
9 | 8 | eqcomi 2830 | . . . . 5 ⊢ {〈𝑂, 𝑥〉} = ({𝑂} × {𝑥}) |
10 | 9 | mpteq2i 5151 | . . . 4 ⊢ (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) = (𝑥 ∈ 𝐾 ↦ ({𝑂} × {𝑥})) |
11 | 10 | mapsnf1o 8497 | . . 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 2822 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐾 = 𝐾) | |
16 | simpr 487 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
17 | xpsng 6896 | . . . . . . . 8 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
18 | 16, 17 | sylancom 590 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) |
19 | 3 | sneqi 4572 | . . . . . . 7 ⊢ {𝑂} = {〈𝐸, 𝐸〉} |
20 | 18, 19 | syl6reqr 2875 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → {𝑂} = ({𝐸} × {𝐸})) |
21 | 20 | oveq2d 7166 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m {𝑂}) = (𝐾 ↑m ({𝐸} × {𝐸}))) |
22 | snfi 8588 | . . . . . 6 ⊢ {𝐸} ∈ Fin | |
23 | simpl 485 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝑅 ∈ Ring) | |
24 | mat1rhmval.a | . . . . . . 7 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
25 | 24, 1 | matbas2 21024 | . . . . . 6 ⊢ (({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑m ({𝐸} × {𝐸})) = (Base‘𝐴)) |
26 | 22, 23, 25 | sylancr 589 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m ({𝐸} × {𝐸})) = (Base‘𝐴)) |
27 | 21, 26 | eqtrd 2856 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m {𝑂}) = (Base‘𝐴)) |
28 | mat1rhmval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
29 | 27, 28 | syl6reqr 2875 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐵 = (𝐾 ↑m {𝑂})) |
30 | 14, 15, 29 | f1oeq123d 6605 | . 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 259 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 {csn 4561 〈cop 4567 ↦ cmpt 5139 × cxp 5548 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 Basecbs 16477 Ringcrg 19291 Mat cmat 21010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-pws 16717 df-sra 19938 df-rgmod 19939 df-dsmm 20870 df-frlm 20885 df-mat 21011 |
This theorem is referenced by: mat1f 21085 mat1rngiso 21089 |
Copyright terms: Public domain | W3C validator |