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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 3264 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) | |
2 | eqcom 2825 | . . . . . 6 ⊢ (𝑋 = 𝑟 ↔ 𝑟 = 𝑋) | |
3 | 2 | rexbii 3244 | . . . . 5 ⊢ (∃𝑟 ∈ 𝐵 𝑋 = 𝑟 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) |
4 | 1, 3 | sylbb2 239 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
5 | 4 | 3ad2ant3 1127 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
6 | mat1dim.o | . . . . . . 7 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
7 | opex 5347 | . . . . . . 7 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
8 | 6, 7 | eqeltri 2906 | . . . . . 6 ⊢ 𝑂 ∈ V |
9 | simp3 1130 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | opthg 5360 | . . . . . 6 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) | |
11 | 8, 9, 10 | sylancr 587 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) |
12 | opex 5347 | . . . . . 6 ⊢ 〈𝑂, 𝑋〉 ∈ V | |
13 | sneqbg 4766 | . . . . . 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 2818 | . . . . . 6 ⊢ 𝑂 = 𝑂 | |
16 | 15 | biantrur 531 | . . . . 5 ⊢ (𝑋 = 𝑟 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟)) |
17 | 11, 14, 16 | 3bitr4g 315 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ 𝑋 = 𝑟)) |
18 | 17 | rexbidv 3294 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ ∃𝑟 ∈ 𝐵 𝑋 = 𝑟)) |
19 | 5, 18 | mpbird 258 | . 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 21008 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
23 | 22 | 3adant3 1124 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
24 | 19, 23 | mpbird 258 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → {〈𝑂, 𝑋〉} ∈ (Base‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 Vcvv 3492 {csn 4557 〈cop 4563 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Ringcrg 19226 Mat cmat 20944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-sra 19873 df-rgmod 19874 df-dsmm 20804 df-frlm 20819 df-mat 20945 |
This theorem is referenced by: mat1dimscm 21012 mat1rhmcl 21018 |
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