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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngplus2-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.p-r | ⊢ + = (+g‘𝐷) |
Ref | Expression |
---|---|
erngplus2-rN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → ((𝑈 + 𝑉)‘𝐹) = ((𝑈‘𝐹) ∘ (𝑉‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erngset.h-r | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | erngset.t-r | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | erngset.e-r | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | erngset.d-r | . . . 4 ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊) | |
5 | erng.p-r | . . . 4 ⊢ + = (+g‘𝐷) | |
6 | 1, 2, 3, 4, 5 | erngplus-rN 40717 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈 + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((𝑈‘𝑓) ∘ (𝑉‘𝑓)))) |
7 | 6 | 3adantr3 1171 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → (𝑈 + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((𝑈‘𝑓) ∘ (𝑉‘𝑓)))) |
8 | fveq2 6919 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑈‘𝑓) = (𝑈‘𝐹)) | |
9 | fveq2 6919 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑉‘𝑓) = (𝑉‘𝐹)) | |
10 | 8, 9 | coeq12d 5888 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑈‘𝑓) ∘ (𝑉‘𝑓)) = ((𝑈‘𝐹) ∘ (𝑉‘𝐹))) |
11 | 10 | adantl 481 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) ∧ 𝑓 = 𝐹) → ((𝑈‘𝑓) ∘ (𝑉‘𝑓)) = ((𝑈‘𝐹) ∘ (𝑉‘𝐹))) |
12 | simpr3 1196 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → 𝐹 ∈ 𝑇) | |
13 | fvex 6932 | . . . 4 ⊢ (𝑈‘𝐹) ∈ V | |
14 | fvex 6932 | . . . 4 ⊢ (𝑉‘𝐹) ∈ V | |
15 | 13, 14 | coex 7966 | . . 3 ⊢ ((𝑈‘𝐹) ∘ (𝑉‘𝐹)) ∈ V |
16 | 15 | a1i 11 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → ((𝑈‘𝐹) ∘ (𝑉‘𝐹)) ∈ V) |
17 | 7, 11, 12, 16 | fvmptd 7034 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → ((𝑈 + 𝑉)‘𝐹) = ((𝑈‘𝐹) ∘ (𝑉‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ↦ cmpt 5252 ∘ ccom 5703 ‘cfv 6572 (class class class)co 7445 +gcplusg 17306 HLchlt 39255 LHypclh 39890 LTrncltrn 40007 TEndoctendo 40658 EDRingRcedring-rN 40660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-struct 17189 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-mulr 17320 df-edring-rN 40662 |
This theorem is referenced by: (None) |
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