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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1scleq | Structured version Visualization version GIF version |
Description: Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
Ref | Expression |
---|---|
ply1scleq.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1scleq.b | ⊢ 𝐵 = (Base‘𝑅) |
ply1scleq.a | ⊢ 𝐴 = (algSc‘𝑃) |
ply1scleq.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ply1scleq.e | ⊢ (𝜑 → 𝐸 ∈ 𝐵) |
ply1scleq.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
ply1scleq | ⊢ (𝜑 → ((𝐴‘𝐸) = (𝐴‘𝐹) ↔ 𝐸 = 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6839 | . . . . 5 ⊢ (𝑑 = 0 → ((coe1‘(𝐴‘𝐸))‘𝑑) = ((coe1‘(𝐴‘𝐸))‘0)) | |
2 | fveq2 6839 | . . . . 5 ⊢ (𝑑 = 0 → ((coe1‘(𝐴‘𝐹))‘𝑑) = ((coe1‘(𝐴‘𝐹))‘0)) | |
3 | 1, 2 | eqeq12d 2752 | . . . 4 ⊢ (𝑑 = 0 → (((coe1‘(𝐴‘𝐸))‘𝑑) = ((coe1‘(𝐴‘𝐹))‘𝑑) ↔ ((coe1‘(𝐴‘𝐸))‘0) = ((coe1‘(𝐴‘𝐹))‘0))) |
4 | ply1scleq.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
5 | ply1scleq.e | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝐵) | |
6 | ply1scleq.p | . . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅) | |
7 | ply1scleq.a | . . . . . . . 8 ⊢ 𝐴 = (algSc‘𝑃) | |
8 | ply1scleq.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
9 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
10 | 6, 7, 8, 9 | ply1sclcl 21641 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝐵) → (𝐴‘𝐸) ∈ (Base‘𝑃)) |
11 | 4, 5, 10 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴‘𝐸) ∈ (Base‘𝑃)) |
12 | ply1scleq.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
13 | 6, 7, 8, 9 | ply1sclcl 21641 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐴‘𝐹) ∈ (Base‘𝑃)) |
14 | 4, 12, 13 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴‘𝐹) ∈ (Base‘𝑃)) |
15 | eqid 2736 | . . . . . . 7 ⊢ (coe1‘(𝐴‘𝐸)) = (coe1‘(𝐴‘𝐸)) | |
16 | eqid 2736 | . . . . . . 7 ⊢ (coe1‘(𝐴‘𝐹)) = (coe1‘(𝐴‘𝐹)) | |
17 | 6, 9, 15, 16 | ply1coe1eq 21653 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐴‘𝐸) ∈ (Base‘𝑃) ∧ (𝐴‘𝐹) ∈ (Base‘𝑃)) → (∀𝑑 ∈ ℕ0 ((coe1‘(𝐴‘𝐸))‘𝑑) = ((coe1‘(𝐴‘𝐹))‘𝑑) ↔ (𝐴‘𝐸) = (𝐴‘𝐹))) |
18 | 4, 11, 14, 17 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → (∀𝑑 ∈ ℕ0 ((coe1‘(𝐴‘𝐸))‘𝑑) = ((coe1‘(𝐴‘𝐹))‘𝑑) ↔ (𝐴‘𝐸) = (𝐴‘𝐹))) |
19 | 18 | biimpar 478 | . . . 4 ⊢ ((𝜑 ∧ (𝐴‘𝐸) = (𝐴‘𝐹)) → ∀𝑑 ∈ ℕ0 ((coe1‘(𝐴‘𝐸))‘𝑑) = ((coe1‘(𝐴‘𝐹))‘𝑑)) |
20 | 0nn0 12424 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ (𝐴‘𝐸) = (𝐴‘𝐹)) → 0 ∈ ℕ0) |
22 | 3, 19, 21 | rspcdva 3580 | . . 3 ⊢ ((𝜑 ∧ (𝐴‘𝐸) = (𝐴‘𝐹)) → ((coe1‘(𝐴‘𝐸))‘0) = ((coe1‘(𝐴‘𝐹))‘0)) |
23 | 6, 7, 8 | ply1sclid 21643 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝐵) → 𝐸 = ((coe1‘(𝐴‘𝐸))‘0)) |
24 | 4, 5, 23 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐸 = ((coe1‘(𝐴‘𝐸))‘0)) |
25 | 24 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐴‘𝐸) = (𝐴‘𝐹)) → 𝐸 = ((coe1‘(𝐴‘𝐸))‘0)) |
26 | 6, 7, 8 | ply1sclid 21643 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 𝐹 = ((coe1‘(𝐴‘𝐹))‘0)) |
27 | 4, 12, 26 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐹 = ((coe1‘(𝐴‘𝐹))‘0)) |
28 | 27 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐴‘𝐸) = (𝐴‘𝐹)) → 𝐹 = ((coe1‘(𝐴‘𝐹))‘0)) |
29 | 22, 25, 28 | 3eqtr4d 2786 | . 2 ⊢ ((𝜑 ∧ (𝐴‘𝐸) = (𝐴‘𝐹)) → 𝐸 = 𝐹) |
30 | fveq2 6839 | . . 3 ⊢ (𝐸 = 𝐹 → (𝐴‘𝐸) = (𝐴‘𝐹)) | |
31 | 30 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝐸 = 𝐹) → (𝐴‘𝐸) = (𝐴‘𝐹)) |
32 | 29, 31 | impbida 799 | 1 ⊢ (𝜑 → ((𝐴‘𝐸) = (𝐴‘𝐹) ↔ 𝐸 = 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ‘cfv 6493 0cc0 11047 ℕ0cn0 12409 Basecbs 17075 Ringcrg 19950 algSccascl 21243 Poly1cpl1 21532 coe1cco1 21533 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-ofr 7614 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-pm 8764 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-fz 13417 df-fzo 13560 df-seq 13899 df-hash 14223 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-sca 17141 df-vsca 17142 df-tset 17144 df-ple 17145 df-0g 17315 df-gsum 17316 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mhm 18593 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-mulg 18864 df-subg 18916 df-ghm 18997 df-cntz 19088 df-cmn 19555 df-abl 19556 df-mgp 19888 df-ur 19905 df-srg 19909 df-ring 19952 df-subrg 20205 df-lmod 20309 df-lss 20378 df-ascl 21246 df-psr 21296 df-mvr 21297 df-mpl 21298 df-opsr 21300 df-psr1 21535 df-vr1 21536 df-ply1 21537 df-coe1 21538 |
This theorem is referenced by: ply1chr 32159 |
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