Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mat1dimbas | Structured version Visualization version GIF version |
Description: A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.) |
Ref | Expression |
---|---|
mat1dim.a | ⊢ 𝐴 = ({𝐸} Mat 𝑅) |
mat1dim.b | ⊢ 𝐵 = (Base‘𝑅) |
mat1dim.o | ⊢ 𝑂 = 〈𝐸, 𝐸〉 |
Ref | Expression |
---|---|
mat1dimbas | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → {〈𝑂, 𝑋〉} ∈ (Base‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 3062 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) | |
2 | eqcom 2629 | . . . . . 6 ⊢ (𝑋 = 𝑟 ↔ 𝑟 = 𝑋) | |
3 | 2 | rexbii 3041 | . . . . 5 ⊢ (∃𝑟 ∈ 𝐵 𝑋 = 𝑟 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) |
4 | 1, 3 | sylbb2 228 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
5 | 4 | 3ad2ant3 1084 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
6 | mat1dim.o | . . . . . . 7 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
7 | opex 4932 | . . . . . . 7 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
8 | 6, 7 | eqeltri 2697 | . . . . . 6 ⊢ 𝑂 ∈ V |
9 | simp3 1063 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | opthg 4946 | . . . . . 6 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) | |
11 | 8, 9, 10 | sylancr 695 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) |
12 | opex 4932 | . . . . . 6 ⊢ 〈𝑂, 𝑋〉 ∈ V | |
13 | sneqbg 4374 | . . . . . 6 ⊢ (〈𝑂, 𝑋〉 ∈ V → ({〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ 〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉)) | |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ ({〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ 〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉) |
15 | eqid 2622 | . . . . . 6 ⊢ 𝑂 = 𝑂 | |
16 | 15 | biantrur 527 | . . . . 5 ⊢ (𝑋 = 𝑟 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟)) |
17 | 11, 14, 16 | 3bitr4g 303 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ 𝑋 = 𝑟)) |
18 | 17 | rexbidv 3052 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ ∃𝑟 ∈ 𝐵 𝑋 = 𝑟)) |
19 | 5, 18 | mpbird 247 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉}) |
20 | mat1dim.a | . . . 4 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
21 | mat1dim.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
22 | 20, 21, 6 | mat1dimelbas 20277 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
23 | 22 | 3adant3 1081 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
24 | 19, 23 | mpbird 247 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → {〈𝑂, 𝑋〉} ∈ (Base‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 {csn 4177 〈cop 4183 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Ringcrg 18547 Mat cmat 20213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-pws 16110 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mat 20214 |
This theorem is referenced by: mat1dimscm 20281 mat1rhmcl 20287 |
Copyright terms: Public domain | W3C validator |