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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngmul-rN | Structured version Visualization version GIF version |
Description: Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
erngset.h-r | ⊢ 𝐻 = (LHyp‘𝐾) |
erngset.t-r | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
erngset.e-r | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
erngset.d-r | ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊) |
erng.m-r | ⊢ · = (.r‘𝐷) |
Ref | Expression |
---|---|
erngmul-rN | ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈 · 𝑉) = (𝑉 ∘ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erngset.h-r | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | erngset.t-r | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | erngset.e-r | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | erngset.d-r | . . . . 5 ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊) | |
5 | erng.m-r | . . . . 5 ⊢ · = (.r‘𝐷) | |
6 | 1, 2, 3, 4, 5 | erngfmul-rN 38564 | . . . 4 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → · = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))) |
7 | 6 | adantr 484 | . . 3 ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → · = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))) |
8 | 7 | oveqd 7230 | . 2 ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈 · 𝑉) = (𝑈(𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))𝑉)) |
9 | coexg 7707 | . . . . 5 ⊢ ((𝑉 ∈ 𝐸 ∧ 𝑈 ∈ 𝐸) → (𝑉 ∘ 𝑈) ∈ V) | |
10 | 9 | ancoms 462 | . . . 4 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑉 ∘ 𝑈) ∈ V) |
11 | coeq2 5727 | . . . . 5 ⊢ (𝑠 = 𝑈 → (𝑡 ∘ 𝑠) = (𝑡 ∘ 𝑈)) | |
12 | coeq1 5726 | . . . . 5 ⊢ (𝑡 = 𝑉 → (𝑡 ∘ 𝑈) = (𝑉 ∘ 𝑈)) | |
13 | eqid 2737 | . . . . 5 ⊢ (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠)) = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠)) | |
14 | 11, 12, 13 | ovmpog 7368 | . . . 4 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (𝑉 ∘ 𝑈) ∈ V) → (𝑈(𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))𝑉) = (𝑉 ∘ 𝑈)) |
15 | 10, 14 | mpd3an3 1464 | . . 3 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈(𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))𝑉) = (𝑉 ∘ 𝑈)) |
16 | 15 | adantl 485 | . 2 ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈(𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))𝑉) = (𝑉 ∘ 𝑈)) |
17 | 8, 16 | eqtrd 2777 | 1 ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈 · 𝑉) = (𝑉 ∘ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∘ ccom 5555 ‘cfv 6380 (class class class)co 7213 ∈ cmpo 7215 .rcmulr 16803 LHypclh 37735 LTrncltrn 37852 TEndoctendo 38503 EDRingRcedring-rN 38505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-mulr 16816 df-edring-rN 38507 |
This theorem is referenced by: erngdvlem3-rN 38749 erngdvlem4-rN 38750 |
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