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 37962 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) → (𝑈 + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((𝑈‘𝑓) ∘ (𝑉‘𝑓)))) |
7 | 6 | 3adantr3 1167 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → (𝑈 + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((𝑈‘𝑓) ∘ (𝑉‘𝑓)))) |
8 | fveq2 6670 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑈‘𝑓) = (𝑈‘𝐹)) | |
9 | fveq2 6670 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑉‘𝑓) = (𝑉‘𝐹)) | |
10 | 8, 9 | coeq12d 5735 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑈‘𝑓) ∘ (𝑉‘𝑓)) = ((𝑈‘𝐹) ∘ (𝑉‘𝐹))) |
11 | 10 | adantl 484 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) ∧ 𝑓 = 𝐹) → ((𝑈‘𝑓) ∘ (𝑉‘𝑓)) = ((𝑈‘𝐹) ∘ (𝑉‘𝐹))) |
12 | simpr3 1192 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → 𝐹 ∈ 𝑇) | |
13 | fvex 6683 | . . . 4 ⊢ (𝑈‘𝐹) ∈ V | |
14 | fvex 6683 | . . . 4 ⊢ (𝑉‘𝐹) ∈ V | |
15 | 13, 14 | coex 7635 | . . 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 6775 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ↦ cmpt 5146 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 +gcplusg 16565 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 TEndoctendo 37903 EDRingRcedring-rN 37905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-edring-rN 37907 |
This theorem is referenced by: (None) |
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