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Mirrors > Home > MPE Home > Th. List > Mathboxes > lineval | Structured version Visualization version GIF version |
Description: A term of the form 𝑥 − 𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉 − 𝐶 (part of ply1remlem 24742). (Contributed by AV, 3-Jul-2019.) |
Ref | Expression |
---|---|
linply1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
linply1.b | ⊢ 𝐵 = (Base‘𝑃) |
linply1.k | ⊢ 𝐾 = (Base‘𝑅) |
linply1.x | ⊢ 𝑋 = (var1‘𝑅) |
linply1.m | ⊢ − = (-g‘𝑃) |
linply1.a | ⊢ 𝐴 = (algSc‘𝑃) |
linply1.g | ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶)) |
linply1.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
lineval.o | ⊢ 𝑂 = (eval1‘𝑅) |
lineval.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
lineval.v | ⊢ (𝜑 → 𝑉 ∈ 𝐾) |
Ref | Expression |
---|---|
lineval | ⊢ (𝜑 → ((𝑂‘𝐺)‘𝑉) = (𝑉(-g‘𝑅)𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | linply1.g | . . . 4 ⊢ 𝐺 = (𝑋 − (𝐴‘𝐶)) | |
2 | 1 | fveq2i 6659 | . . 3 ⊢ (𝑂‘𝐺) = (𝑂‘(𝑋 − (𝐴‘𝐶))) |
3 | 2 | fveq1i 6657 | . 2 ⊢ ((𝑂‘𝐺)‘𝑉) = ((𝑂‘(𝑋 − (𝐴‘𝐶)))‘𝑉) |
4 | lineval.o | . . . 4 ⊢ 𝑂 = (eval1‘𝑅) | |
5 | linply1.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | linply1.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
7 | linply1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
8 | lineval.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
9 | lineval.v | . . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐾) | |
10 | linply1.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
11 | 4, 10, 6, 5, 7, 8, 9 | evl1vard 20483 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑉) = 𝑉)) |
12 | linply1.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
13 | linply1.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
14 | 4, 5, 6, 12, 7, 8, 13, 9 | evl1scad 20481 | . . . 4 ⊢ (𝜑 → ((𝐴‘𝐶) ∈ 𝐵 ∧ ((𝑂‘(𝐴‘𝐶))‘𝑉) = 𝐶)) |
15 | linply1.m | . . . 4 ⊢ − = (-g‘𝑃) | |
16 | eqid 2821 | . . . 4 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
17 | 4, 5, 6, 7, 8, 9, 11, 14, 15, 16 | evl1subd 20488 | . . 3 ⊢ (𝜑 → ((𝑋 − (𝐴‘𝐶)) ∈ 𝐵 ∧ ((𝑂‘(𝑋 − (𝐴‘𝐶)))‘𝑉) = (𝑉(-g‘𝑅)𝐶))) |
18 | 17 | simprd 498 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑋 − (𝐴‘𝐶)))‘𝑉) = (𝑉(-g‘𝑅)𝐶)) |
19 | 3, 18 | syl5eq 2868 | 1 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝑉) = (𝑉(-g‘𝑅)𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 -gcsg 18088 CRingccrg 19281 algSccascl 20067 var1cv1 20327 Poly1cpl1 20328 eval1ce1 20460 |
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 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
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-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-ofr 7396 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-sup 8892 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-fz 12883 df-fzo 13024 df-seq 13360 df-hash 13681 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-hom 16572 df-cco 16573 df-0g 16698 df-gsum 16699 df-prds 16704 df-pws 16706 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-mhm 17939 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-mulg 18208 df-subg 18259 df-ghm 18339 df-cntz 18430 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-srg 19239 df-ring 19282 df-cring 19283 df-rnghom 19450 df-subrg 19516 df-lmod 19619 df-lss 19687 df-lsp 19727 df-assa 20068 df-asp 20069 df-ascl 20070 df-psr 20119 df-mvr 20120 df-mpl 20121 df-opsr 20123 df-evls 20269 df-evl 20270 df-psr1 20331 df-vr1 20332 df-ply1 20333 df-evl1 20462 |
This theorem is referenced by: linevalexample 44535 |
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