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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 3267 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) | |
2 | eqcom 2828 | . . . . . 6 ⊢ (𝑋 = 𝑟 ↔ 𝑟 = 𝑋) | |
3 | 2 | rexbii 3247 | . . . . 5 ⊢ (∃𝑟 ∈ 𝐵 𝑋 = 𝑟 ↔ ∃𝑟 ∈ 𝐵 𝑟 = 𝑋) |
4 | 1, 3 | sylbb2 240 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
5 | 4 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ∃𝑟 ∈ 𝐵 𝑋 = 𝑟) |
6 | mat1dim.o | . . . . . . 7 ⊢ 𝑂 = 〈𝐸, 𝐸〉 | |
7 | opex 5356 | . . . . . . 7 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
8 | 6, 7 | eqeltri 2909 | . . . . . 6 ⊢ 𝑂 ∈ V |
9 | simp3 1134 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | opthg 5369 | . . . . . 6 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) | |
11 | 8, 9, 10 | sylancr 589 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (〈𝑂, 𝑋〉 = 〈𝑂, 𝑟〉 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟))) |
12 | opex 5356 | . . . . . 6 ⊢ 〈𝑂, 𝑋〉 ∈ V | |
13 | sneqbg 4774 | . . . . . 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 2821 | . . . . . 6 ⊢ 𝑂 = 𝑂 | |
16 | 15 | biantrur 533 | . . . . 5 ⊢ (𝑋 = 𝑟 ↔ (𝑂 = 𝑂 ∧ 𝑋 = 𝑟)) |
17 | 11, 14, 16 | 3bitr4g 316 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ 𝑋 = 𝑟)) |
18 | 17 | rexbidv 3297 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉} ↔ ∃𝑟 ∈ 𝐵 𝑋 = 𝑟)) |
19 | 5, 18 | mpbird 259 | . 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 21080 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
23 | 22 | 3adant3 1128 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ({〈𝑂, 𝑋〉} ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 {〈𝑂, 𝑋〉} = {〈𝑂, 𝑟〉})) |
24 | 19, 23 | mpbird 259 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → {〈𝑂, 𝑋〉} ∈ (Base‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 {csn 4567 〈cop 4573 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Ringcrg 19297 Mat cmat 21016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-sra 19944 df-rgmod 19945 df-dsmm 20876 df-frlm 20891 df-mat 21017 |
This theorem is referenced by: mat1dimscm 21084 mat1rhmcl 21090 |
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