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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 20206 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
5 | mamumat1cl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 20207 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
7 | 4, 6 | ifcld 4570 | . . . . . 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 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑀)) → if(𝑖 = 𝑗, 1 , 0 ) ∈ 𝐵) |
10 | 9 | ralrimivva 3191 | . . 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 8070 | . . 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 6906 | . . 3 ⊢ 𝐵 ∈ V |
15 | mamumat1cl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
16 | xpfi 9341 | . . . 4 ⊢ ((𝑀 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑀 × 𝑀) ∈ Fin) | |
17 | 15, 15, 16 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑀 × 𝑀) ∈ Fin) |
18 | elmapg 8856 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑀) ∈ Fin) → (𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀)) ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵)) | |
19 | 14, 17, 18 | sylancr 585 | . 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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ifcif 4524 × cxp 5670 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 ∈ cmpo 7418 ↑m cmap 8843 Fincfn 8962 Basecbs 17179 0gc0g 17420 1rcur 20125 Ringcrg 20177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-mgp 20079 df-ur 20126 df-ring 20179 |
This theorem is referenced by: mamulid 22361 mamurid 22362 matring 22363 mat1 22367 |
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