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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 20165 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
5 | mamumat1cl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | 2, 5 | ring0cl 20166 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
7 | 4, 6 | ifcld 4569 | . . . . . 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 3194 | . . 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 8053 | . . 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 6899 | . . 3 ⊢ 𝐵 ∈ V |
15 | mamumat1cl.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
16 | xpfi 9319 | . . . 4 ⊢ ((𝑀 ∈ Fin ∧ 𝑀 ∈ Fin) → (𝑀 × 𝑀) ∈ Fin) | |
17 | 15, 15, 16 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑀 × 𝑀) ∈ Fin) |
18 | elmapg 8835 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑀) ∈ Fin) → (𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀)) ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵)) | |
19 | 14, 17, 18 | sylancr 586 | . 2 ⊢ (𝜑 → (𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀)) ↔ 𝐼:(𝑀 × 𝑀)⟶𝐵)) |
20 | 13, 19 | mpbird 257 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ifcif 4523 × cxp 5667 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 ↑m cmap 8822 Fincfn 8941 Basecbs 17153 0gc0g 17394 1rcur 20086 Ringcrg 20138 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-mgp 20040 df-ur 20087 df-ring 20140 |
This theorem is referenced by: mamulid 22298 mamurid 22299 matring 22300 mat1 22304 |
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