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Mirrors > Home > MPE Home > Th. List > pf1const | Structured version Visualization version GIF version |
Description: Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
pf1const.b | ⊢ 𝐵 = (Base‘𝑅) |
pf1const.q | ⊢ 𝑄 = ran (eval1‘𝑅) |
Ref | Expression |
---|---|
pf1const | ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐵 × {𝑋}) ∈ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ (eval1‘𝑅) = (eval1‘𝑅) | |
2 | eqid 2734 | . . . 4 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
3 | pf1const.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | eqid 2734 | . . . 4 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅)) | |
5 | 1, 2, 3, 4 | evl1sca 22352 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑋)) = (𝐵 × {𝑋})) |
6 | eqid 2734 | . . . . . . 7 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
7 | 1, 2, 6, 3 | evl1rhm 22350 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵))) |
9 | eqid 2734 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
10 | eqid 2734 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
11 | 9, 10 | rhmf 20506 | . . . . 5 ⊢ ((eval1‘𝑅) ∈ ((Poly1‘𝑅) RingHom (𝑅 ↑s 𝐵)) → (eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵))) |
12 | ffn 6746 | . . . . 5 ⊢ ((eval1‘𝑅):(Base‘(Poly1‘𝑅))⟶(Base‘(𝑅 ↑s 𝐵)) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅))) | |
13 | 8, 11, 12 | 3syl 18 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (eval1‘𝑅) Fn (Base‘(Poly1‘𝑅))) |
14 | crngring 20267 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
15 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Ring) |
16 | 2, 4, 3, 9 | ply1sclf 22302 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (algSc‘(Poly1‘𝑅)):𝐵⟶(Base‘(Poly1‘𝑅))) |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (algSc‘(Poly1‘𝑅)):𝐵⟶(Base‘(Poly1‘𝑅))) |
18 | ffvelcdm 7113 | . . . . 5 ⊢ (((algSc‘(Poly1‘𝑅)):𝐵⟶(Base‘(Poly1‘𝑅)) ∧ 𝑋 ∈ 𝐵) → ((algSc‘(Poly1‘𝑅))‘𝑋) ∈ (Base‘(Poly1‘𝑅))) | |
19 | 17, 18 | sylancom 587 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((algSc‘(Poly1‘𝑅))‘𝑋) ∈ (Base‘(Poly1‘𝑅))) |
20 | fnfvelrn 7112 | . . . 4 ⊢ (((eval1‘𝑅) Fn (Base‘(Poly1‘𝑅)) ∧ ((algSc‘(Poly1‘𝑅))‘𝑋) ∈ (Base‘(Poly1‘𝑅))) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑋)) ∈ ran (eval1‘𝑅)) | |
21 | 13, 19, 20 | syl2anc 583 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑋)) ∈ ran (eval1‘𝑅)) |
22 | 5, 21 | eqeltrrd 2839 | . 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐵 × {𝑋}) ∈ ran (eval1‘𝑅)) |
23 | pf1const.q | . 2 ⊢ 𝑄 = ran (eval1‘𝑅) | |
24 | 22, 23 | eleqtrrdi 2849 | 1 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵) → (𝐵 × {𝑋}) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 {csn 4648 × cxp 5697 ran crn 5700 Fn wfn 6567 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 ↑s cpws 17501 Ringcrg 20255 CRingccrg 20256 RingHom crh 20490 algSccascl 21890 Poly1cpl1 22192 eval1ce1 22332 |
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-rmo 3383 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-int 4973 df-iun 5021 df-iin 5022 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-se 5655 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-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-ofr 7711 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-sup 9507 df-oi 9575 df-card 10004 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-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-fzo 13708 df-seq 14049 df-hash 14376 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-hom 17330 df-cco 17331 df-0g 17496 df-gsum 17497 df-prds 17502 df-pws 17504 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-mulg 19103 df-subg 19158 df-ghm 19248 df-cntz 19352 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-srg 20209 df-ring 20257 df-cring 20258 df-rhm 20493 df-subrng 20567 df-subrg 20592 df-lmod 20877 df-lss 20948 df-lsp 20988 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21946 df-mvr 21947 df-mpl 21948 df-opsr 21950 df-evls 22115 df-evl 22116 df-psr1 22195 df-ply1 22197 df-evl1 22334 |
This theorem is referenced by: (None) |
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