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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 6920 | . . . 4 ⊢ 𝐾 ∈ V |
| 3 | mat1rhmval.o | . . . . 5 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
| 4 | opex 5469 | . . . . 5 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
| 5 | 3, 4 | eqeltri 2837 | . . . 4 ⊢ 𝑂 ∈ V |
| 6 | 2, 5 | pm3.2i 470 | . . 3 ⊢ (𝐾 ∈ V ∧ 𝑂 ∈ V) |
| 7 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 8 | 5, 7 | xpsn 7161 | . . . . . 6 ⊢ ({𝑂} × {𝑥}) = {〈𝑂, 𝑥〉} |
| 9 | 8 | eqcomi 2746 | . . . . 5 ⊢ {〈𝑂, 𝑥〉} = ({𝑂} × {𝑥}) |
| 10 | 9 | mpteq2i 5247 | . . . 4 ⊢ (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) = (𝑥 ∈ 𝐾 ↦ ({𝑂} × {𝑥})) |
| 11 | 10 | mapsnf1o 8979 | . . 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 4637 | . . . . . . 7 ⊢ {𝑂} = {〈𝐸, 𝐸〉} |
| 18 | simpr 484 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
| 19 | xpsng 7159 | . . . . . . . 8 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
| 20 | 18, 19 | sylancom 588 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) |
| 21 | 17, 20 | eqtr4id 2796 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → {𝑂} = ({𝐸} × {𝐸})) |
| 22 | 21 | oveq2d 7447 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → (𝐾 ↑m {𝑂}) = (𝐾 ↑m ({𝐸} × {𝐸}))) |
| 23 | snfi 9083 | . . . . . 6 ⊢ {𝐸} ∈ Fin | |
| 24 | simpl 482 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝑅 ∈ Ring) | |
| 25 | mat1rhmval.a | . . . . . . 7 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
| 26 | 25, 1 | matbas2 22427 | . . . . . 6 ⊢ (({𝐸} ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑m ({𝐸} × {𝐸})) = (Base‘𝐴)) |
| 27 | 23, 24, 26 | sylancr 587 | . . . . 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 2796 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐵 = (𝐾 ↑m {𝑂})) |
| 30 | 14, 15, 29 | f1oeq123d 6842 | . 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 257 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ↦ cmpt 5225 × cxp 5683 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 Basecbs 17247 Ringcrg 20230 Mat cmat 22411 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-mat 22412 |
| This theorem is referenced by: mat1f 22488 mat1rngiso 22492 |
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