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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 3226 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) | |
2 | eqcom 2805 | . . . . . 6 ⊢ (𝑋 = 𝑟 ↔ 𝑟 = 𝑋) | |
3 | 2 | rexbii 3210 | . . . . 5 ⊢ (∃𝑟 ∈ 𝐵 𝑋 = 𝑟 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) |
4 | 1, 3 | sylbb2 241 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
5 | 4 | 3ad2ant3 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
6 | mat1dim.o | . . . . . . 7 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
7 | opex 5321 | . . . . . . 7 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
8 | 6, 7 | eqeltri 2886 | . . . . . 6 ⊢ 𝑂 ∈ V |
9 | simp3 1135 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | opthg 5334 | . . . . . 6 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) | |
11 | 8, 9, 10 | sylancr 590 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) |
12 | opex 5321 | . . . . . 6 ⊢ 〈𝑂, 𝑋〉 ∈ V | |
13 | sneqbg 4734 | . . . . . 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 2798 | . . . . . 6 ⊢ 𝑂 = 𝑂 | |
16 | 15 | biantrur 534 | . . . . 5 ⊢ (𝑋 = 𝑟 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟)) |
17 | 11, 14, 16 | 3bitr4g 317 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ 𝑋 = 𝑟)) |
18 | 17 | rexbidv 3256 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ ∃𝑟 ∈ 𝐵 𝑋 = 𝑟)) |
19 | 5, 18 | mpbird 260 | . 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 21076 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
23 | 22 | 3adant3 1129 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
24 | 19, 23 | mpbird 260 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → {〈𝑂, 𝑋〉} ∈ (Base‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 {csn 4525 〈cop 4531 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Ringcrg 19290 Mat cmat 21012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-sra 19937 df-rgmod 19938 df-dsmm 20421 df-frlm 20436 df-mat 21013 |
This theorem is referenced by: mat1dimscm 21080 mat1rhmcl 21086 |
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