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Mirrors > Home > MPE Home > Th. List > mamumat1cl | Structured version Visualization version GIF version |
Description: The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
Ref | Expression |
---|---|
mamumat1cl.b | ⊢ 𝐵 = (Base‘𝑅) |
mamumat1cl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mamumat1cl.o | ⊢ 1 = (1r‘𝑅) |
mamumat1cl.z | ⊢ 0 = (0g‘𝑅) |
mamumat1cl.i | ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) |
mamumat1cl.m | ⊢ (𝜑 → 𝑀 ∈ Fin) |
Ref | Expression |
---|---|
mamumat1cl | ⊢ (𝜑 → 𝐼 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mamumat1cl.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | mamumat1cl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
3 | mamumat1cl.o | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 18614 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
5 | mamumat1cl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 18615 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
7 | 4, 6 | ifcld 4164 | . . . . . 6 ⊢ (𝑅 ∈ Ring → if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵) |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑀)) → if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵) |
10 | 9 | ralrimivva 3000 | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑀 ∀𝑗 ∈ 𝑀 if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵) |
11 | mamumat1cl.i | . . . 4 ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) | |
12 | 11 | fmpt2 7282 | . . 3 ⊢ (∀𝑖 ∈ 𝑀 ∀𝑗 ∈ 𝑀 if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵 ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵) |
13 | 10, 12 | sylib 208 | . 2 ⊢ (𝜑 → 𝐼:(𝑀 × 𝑀)⟶𝐵) |
14 | fvex 6239 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
15 | 2, 14 | eqeltri 2726 | . . 3 ⊢ 𝐵 ∈ V |
16 | mamumat1cl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
17 | xpfi 8272 | . . . 4 ⊢ ((𝑀 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑀 × 𝑀) ∈ Fin) | |
18 | 16, 16, 17 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝑀 × 𝑀) ∈ Fin) |
19 | elmapg 7912 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑀) ∈ Fin) → (𝐼 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑀)) ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵)) | |
20 | 15, 18, 19 | sylancr 696 | . 2 ⊢ (𝜑 → (𝐼 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑀)) ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵)) |
21 | 13, 20 | mpbird 247 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 ifcif 4119 × cxp 5141 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 ↑𝑚 cmap 7899 Fincfn 7997 Basecbs 15904 0gc0g 16147 1rcur 18547 Ringcrg 18593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-mgp 18536 df-ur 18548 df-ring 18595 |
This theorem is referenced by: mamulid 20295 mamurid 20296 matring 20297 mat1 20301 |
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