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Mirrors > Home > MPE Home > Th. List > mon1pval | Structured version Visualization version Unicode version |
Description: Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pval.p | Poly1 |
uc1pval.b | |
uc1pval.z | |
uc1pval.d | deg1 |
mon1pval.m | Monic1p |
mon1pval.o |
Ref | Expression |
---|---|
mon1pval | coe1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mon1pval.m | . 2 Monic1p | |
2 | fveq2 6191 | . . . . . . . 8 Poly1 Poly1 | |
3 | uc1pval.p | . . . . . . . 8 Poly1 | |
4 | 2, 3 | syl6eqr 2674 | . . . . . . 7 Poly1 |
5 | 4 | fveq2d 6195 | . . . . . 6 Poly1 |
6 | uc1pval.b | . . . . . 6 | |
7 | 5, 6 | syl6eqr 2674 | . . . . 5 Poly1 |
8 | 4 | fveq2d 6195 | . . . . . . . 8 Poly1 |
9 | uc1pval.z | . . . . . . . 8 | |
10 | 8, 9 | syl6eqr 2674 | . . . . . . 7 Poly1 |
11 | 10 | neeq2d 2854 | . . . . . 6 Poly1 |
12 | fveq2 6191 | . . . . . . . . . 10 deg1 deg1 | |
13 | uc1pval.d | . . . . . . . . . 10 deg1 | |
14 | 12, 13 | syl6eqr 2674 | . . . . . . . . 9 deg1 |
15 | 14 | fveq1d 6193 | . . . . . . . 8 deg1 |
16 | 15 | fveq2d 6195 | . . . . . . 7 coe1 deg1 coe1 |
17 | fveq2 6191 | . . . . . . . 8 | |
18 | mon1pval.o | . . . . . . . 8 | |
19 | 17, 18 | syl6eqr 2674 | . . . . . . 7 |
20 | 16, 19 | eqeq12d 2637 | . . . . . 6 coe1 deg1 coe1 |
21 | 11, 20 | anbi12d 747 | . . . . 5 Poly1 coe1 deg1 coe1 |
22 | 7, 21 | rabeqbidv 3195 | . . . 4 Poly1 Poly1 coe1 deg1 coe1 |
23 | df-mon1 23890 | . . . 4 Monic1p Poly1 Poly1 coe1 deg1 | |
24 | fvex 6201 | . . . . . 6 | |
25 | 6, 24 | eqeltri 2697 | . . . . 5 |
26 | 25 | rabex 4813 | . . . 4 coe1 |
27 | 22, 23, 26 | fvmpt 6282 | . . 3 Monic1p coe1 |
28 | fvprc 6185 | . . . 4 Monic1p | |
29 | ssrab2 3687 | . . . . . 6 coe1 | |
30 | fvprc 6185 | . . . . . . . . . 10 Poly1 | |
31 | 3, 30 | syl5eq 2668 | . . . . . . . . 9 |
32 | 31 | fveq2d 6195 | . . . . . . . 8 |
33 | 6, 32 | syl5eq 2668 | . . . . . . 7 |
34 | base0 15912 | . . . . . . 7 | |
35 | 33, 34 | syl6eqr 2674 | . . . . . 6 |
36 | 29, 35 | syl5sseq 3653 | . . . . 5 coe1 |
37 | ss0 3974 | . . . . 5 coe1 coe1 | |
38 | 36, 37 | syl 17 | . . . 4 coe1 |
39 | 28, 38 | eqtr4d 2659 | . . 3 Monic1p coe1 |
40 | 27, 39 | pm2.61i 176 | . 2 Monic1p coe1 |
41 | 1, 40 | eqtri 2644 | 1 coe1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wa 384 wceq 1483 wcel 1990 wne 2794 crab 2916 cvv 3200 wss 3574 c0 3915 cfv 5888 cbs 15857 c0g 16100 cur 18501 Poly1cpl1 19547 coe1cco1 19548 deg1 cdg1 23814 Monic1pcmn1 23885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-slot 15861 df-base 15863 df-mon1 23890 |
This theorem is referenced by: ismon1p 23902 |
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