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Mirrors > Home > MPE Home > Th. List > deg1propd | Structured version Visualization version GIF version |
Description: Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
deg1propd.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
deg1propd.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) |
deg1propd.p | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) |
Ref | Expression |
---|---|
deg1propd | ⊢ (𝜑 → ( deg1 ‘𝑅) = ( deg1 ‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1propd.b1 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
2 | deg1propd.b2 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) | |
3 | deg1propd.p | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) | |
4 | 1, 2, 3 | mdegpropd 25401 | . 2 ⊢ (𝜑 → (1o mDeg 𝑅) = (1o mDeg 𝑆)) |
5 | eqid 2737 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
6 | 5 | deg1fval 25397 | . 2 ⊢ ( deg1 ‘𝑅) = (1o mDeg 𝑅) |
7 | eqid 2737 | . . 3 ⊢ ( deg1 ‘𝑆) = ( deg1 ‘𝑆) | |
8 | 7 | deg1fval 25397 | . 2 ⊢ ( deg1 ‘𝑆) = (1o mDeg 𝑆) |
9 | 4, 6, 8 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → ( deg1 ‘𝑅) = ( deg1 ‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 1oc1o 8397 Basecbs 17043 +gcplusg 17093 mDeg cmdg 25367 deg1 cdg1 25368 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 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-iun 4954 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-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-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-tset 17112 df-0g 17283 df-psr 21264 df-mpl 21266 df-mdeg 25369 df-deg1 25370 |
This theorem is referenced by: ply1divalg2 25455 |
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