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Mirrors > Home > MPE Home > Th. List > mdet0f1o | Structured version Visualization version GIF version |
Description: The determinant function for 0-dimensional matrices on a given ring is a bijection from the singleton containing the empty set (empty matrix) onto the singleton containing the unity element of that ring. (Contributed by AV, 28-Feb-2019.) |
Ref | Expression |
---|---|
mdet0f1o | ⊢ (𝑅 ∈ Ring → (∅ maDet 𝑅):{∅}–1-1-onto→{(1r‘𝑅)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdet0pr 21957 | . 2 ⊢ (𝑅 ∈ Ring → (∅ maDet 𝑅) = {⟨∅, (1r‘𝑅)⟩}) | |
2 | 0ex 5265 | . . . 4 ⊢ ∅ ∈ V | |
3 | fvex 6856 | . . . 4 ⊢ (1r‘𝑅) ∈ V | |
4 | 2, 3 | f1osn 6825 | . . 3 ⊢ {⟨∅, (1r‘𝑅)⟩}:{∅}–1-1-onto→{(1r‘𝑅)} |
5 | f1oeq1 6773 | . . 3 ⊢ ((∅ maDet 𝑅) = {⟨∅, (1r‘𝑅)⟩} → ((∅ maDet 𝑅):{∅}–1-1-onto→{(1r‘𝑅)} ↔ {⟨∅, (1r‘𝑅)⟩}:{∅}–1-1-onto→{(1r‘𝑅)})) | |
6 | 4, 5 | mpbiri 258 | . 2 ⊢ ((∅ maDet 𝑅) = {⟨∅, (1r‘𝑅)⟩} → (∅ maDet 𝑅):{∅}–1-1-onto→{(1r‘𝑅)}) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝑅 ∈ Ring → (∅ maDet 𝑅):{∅}–1-1-onto→{(1r‘𝑅)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∅c0 4283 {csn 4587 ⟨cop 4593 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 1rcur 19918 Ringcrg 19969 maDet cmdat 21949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-xnn0 12491 df-z 12505 df-dec 12624 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-word 14409 df-lsw 14457 df-concat 14465 df-s1 14490 df-substr 14535 df-pfx 14565 df-splice 14644 df-reverse 14653 df-s2 14743 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-efmnd 18684 df-grp 18756 df-minusg 18757 df-mulg 18878 df-subg 18930 df-ghm 19011 df-gim 19054 df-cntz 19102 df-oppg 19129 df-symg 19154 df-pmtr 19229 df-psgn 19278 df-evpm 19279 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-rnghom 20153 df-drng 20199 df-subrg 20234 df-sra 20649 df-rgmod 20650 df-cnfld 20813 df-zring 20886 df-zrh 20920 df-dsmm 21154 df-frlm 21169 df-mat 21771 df-mdet 21950 |
This theorem is referenced by: (None) |
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