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Mirrors > Home > MPE Home > Th. List > evls1rhmlem | Structured version Visualization version GIF version |
Description: Lemma for evl1rhm 22225 and evls1rhm 22215 (formerly part of the proof of evl1rhm 22225): The first function of the composition forming the univariate polynomial evaluation map function for a (sub)ring is a ring homomorphism. (Contributed by AV, 11-Sep-2019.) |
Ref | Expression |
---|---|
evl1rhmlem.b | ⊢ 𝐵 = (Base‘𝑅) |
evl1rhmlem.t | ⊢ 𝑇 = (𝑅 ↑s 𝐵) |
evl1rhmlem.f | ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
Ref | Expression |
---|---|
evls1rhmlem | ⊢ (𝑅 ∈ CRing → 𝐹 ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1rhmlem.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | |
2 | ovex 7447 | . . . . 5 ⊢ (𝐵 ↑m 1o) ∈ V | |
3 | eqid 2727 | . . . . . 6 ⊢ (𝑅 ↑s (𝐵 ↑m 1o)) = (𝑅 ↑s (𝐵 ↑m 1o)) | |
4 | evl1rhmlem.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | pwsbas 17454 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ↑m 1o) ∈ V) → (𝐵 ↑m (𝐵 ↑m 1o)) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
6 | 2, 5 | mpan2 690 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝐵 ↑m (𝐵 ↑m 1o)) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o)))) |
7 | 6 | mpteq1d 5237 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) = (𝑥 ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))) |
8 | 1, 7 | eqtrid 2779 | . 2 ⊢ (𝑅 ∈ CRing → 𝐹 = (𝑥 ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))) |
9 | evl1rhmlem.t | . . 3 ⊢ 𝑇 = (𝑅 ↑s 𝐵) | |
10 | eqid 2727 | . . 3 ⊢ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) = (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) | |
11 | crngring 20169 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
12 | 4 | fvexi 6905 | . . . 4 ⊢ 𝐵 ∈ V |
13 | 12 | a1i 11 | . . 3 ⊢ (𝑅 ∈ CRing → 𝐵 ∈ V) |
14 | 2 | a1i 11 | . . 3 ⊢ (𝑅 ∈ CRing → (𝐵 ↑m 1o) ∈ V) |
15 | df1o2 8485 | . . . . 5 ⊢ 1o = {∅} | |
16 | 0ex 5301 | . . . . 5 ⊢ ∅ ∈ V | |
17 | eqid 2727 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) | |
18 | 15, 12, 16, 17 | mapsnf1o3 8903 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑m 1o) |
19 | f1of 6833 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵–1-1-onto→(𝐵 ↑m 1o) → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) | |
20 | 18, 19 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})):𝐵⟶(𝐵 ↑m 1o)) |
21 | 9, 3, 10, 11, 13, 14, 20 | pwsco1rhm 20423 | . 2 ⊢ (𝑅 ∈ CRing → (𝑥 ∈ (Base‘(𝑅 ↑s (𝐵 ↑m 1o))) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
22 | 8, 21 | eqeltrd 2828 | 1 ⊢ (𝑅 ∈ CRing → 𝐹 ∈ ((𝑅 ↑s (𝐵 ↑m 1o)) RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∅c0 4318 {csn 4624 ↦ cmpt 5225 × cxp 5670 ∘ ccom 5676 ⟶wf 6538 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 1oc1o 8471 ↑m cmap 8834 Basecbs 17165 ↑s cpws 17413 CRingccrg 20158 RingHom crh 20390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-prds 17414 df-pws 17416 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-grp 18878 df-minusg 18879 df-ghm 19152 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-rhm 20393 |
This theorem is referenced by: evls1rhm 22215 evl1rhm 22225 |
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