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Mirrors > Home > MPE Home > Th. List > ismon1p | Structured version Visualization version GIF version |
Description: Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
uc1pval.b | ⊢ 𝐵 = (Base‘𝑃) |
uc1pval.z | ⊢ 0 = (0g‘𝑃) |
uc1pval.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
mon1pval.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
mon1pval.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ismon1p | ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 3003 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 )) | |
2 | fveq2 6879 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹)) | |
3 | fveq2 6879 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹)) | |
4 | 2, 3 | fveq12d 6886 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹))) |
5 | 4 | eqeq1d 2734 | . . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
6 | 1, 5 | anbi12d 631 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ) ↔ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) |
7 | uc1pval.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
8 | uc1pval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
9 | uc1pval.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
10 | uc1pval.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
11 | mon1pval.m | . . . 4 ⊢ 𝑀 = (Monic1p‘𝑅) | |
12 | mon1pval.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
13 | 7, 8, 9, 10, 11, 12 | mon1pval 25590 | . . 3 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} |
14 | 6, 13 | elrab2 3683 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) |
15 | 3anass 1095 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))) | |
16 | 14, 15 | bitr4i 277 | 1 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ‘cfv 6533 Basecbs 17128 0gc0g 17369 1rcur 19965 Poly1cpl1 21632 coe1cco1 21633 deg1 cdg1 25500 Monic1pcmn1 25574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-1cn 11152 ax-addcl 11154 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-nn 12197 df-slot 17099 df-ndx 17111 df-base 17129 df-mon1 25579 |
This theorem is referenced by: mon1pcl 25593 mon1pn0 25595 mon1pldg 25598 uc1pmon1p 25600 ply1remlem 25611 0ringmon1p 32545 ressply1mon1p 32561 mon1pid 41782 mon1psubm 41783 |
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