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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 19803 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
5 | mamumat1cl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 19804 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
7 | 4, 6 | ifcld 4511 | . . . . . 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 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑀)) → if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵) |
10 | 9 | ralrimivva 3117 | . . 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 | fmpo 7899 | . . 3 ⊢ (∀𝑖 ∈ 𝑀 ∀𝑗 ∈ 𝑀 if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵 ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵) |
13 | 10, 12 | sylib 217 | . 2 ⊢ (𝜑 → 𝐼:(𝑀 × 𝑀)⟶𝐵) |
14 | 2 | fvexi 6783 | . . 3 ⊢ 𝐵 ∈ V |
15 | mamumat1cl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
16 | xpfi 9061 | . . . 4 ⊢ ((𝑀 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑀 × 𝑀) ∈ Fin) | |
17 | 15, 15, 16 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑀 × 𝑀) ∈ Fin) |
18 | elmapg 8609 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑀) ∈ Fin) → (𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀)) ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵)) | |
19 | 14, 17, 18 | sylancr 587 | . 2 ⊢ (𝜑 → (𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀)) ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵)) |
20 | 13, 19 | mpbird 256 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 Vcvv 3431 ifcif 4465 × cxp 5587 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 ∈ cmpo 7271 ↑m cmap 8596 Fincfn 8714 Basecbs 16908 0gc0g 17146 1rcur 19733 Ringcrg 19779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-0g 17148 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-grp 18576 df-mgp 19717 df-ur 19734 df-ring 19781 |
This theorem is referenced by: mamulid 21586 mamurid 21587 matring 21588 mat1 21592 |
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