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Mirrors > Home > MPE Home > Th. List > mdegpropd | Structured version Visualization version GIF version |
Description: Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
Ref | Expression |
---|---|
mdegpropd.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
mdegpropd.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) |
mdegpropd.p | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) |
Ref | Expression |
---|---|
mdegpropd | ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdegpropd.b1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
2 | mdegpropd.b2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) | |
3 | mdegpropd.p | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) | |
4 | 1, 2, 3 | mplbaspropd 20407 | . . 3 ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
5 | 1, 2, 3 | grpidpropd 17874 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝑆)) |
6 | 5 | oveq2d 7174 | . . . . 5 ⊢ (𝜑 → (𝑐 supp (0g‘𝑅)) = (𝑐 supp (0g‘𝑆))) |
7 | 6 | imaeq2d 5931 | . . . 4 ⊢ (𝜑 → ((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))) = ((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆)))) |
8 | 7 | supeq1d 8912 | . . 3 ⊢ (𝜑 → sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))), ℝ*, < ) = sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆))), ℝ*, < )) |
9 | 4, 8 | mpteq12dv 5153 | . 2 ⊢ (𝜑 → (𝑐 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))), ℝ*, < )) = (𝑐 ∈ (Base‘(𝐼 mPoly 𝑆)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆))), ℝ*, < ))) |
10 | eqid 2823 | . . 3 ⊢ (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑅) | |
11 | eqid 2823 | . . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
12 | eqid 2823 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
13 | eqid 2823 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | eqid 2823 | . . 3 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
15 | eqid 2823 | . . 3 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) = (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) | |
16 | 10, 11, 12, 13, 14, 15 | mdegfval 24658 | . 2 ⊢ (𝐼 mDeg 𝑅) = (𝑐 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑅))), ℝ*, < )) |
17 | eqid 2823 | . . 3 ⊢ (𝐼 mDeg 𝑆) = (𝐼 mDeg 𝑆) | |
18 | eqid 2823 | . . 3 ⊢ (𝐼 mPoly 𝑆) = (𝐼 mPoly 𝑆) | |
19 | eqid 2823 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑆)) = (Base‘(𝐼 mPoly 𝑆)) | |
20 | eqid 2823 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
21 | 17, 18, 19, 20, 14, 15 | mdegfval 24658 | . 2 ⊢ (𝐼 mDeg 𝑆) = (𝑐 ∈ (Base‘(𝐼 mPoly 𝑆)) ↦ sup(((𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑏)) “ (𝑐 supp (0g‘𝑆))), ℝ*, < )) |
22 | 9, 16, 21 | 3eqtr4g 2883 | 1 ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 ↦ cmpt 5148 ◡ccnv 5556 “ cima 5560 ‘cfv 6357 (class class class)co 7158 supp csupp 7832 ↑m cmap 8408 Fincfn 8511 supcsup 8906 ℝ*cxr 10676 < clt 10677 ℕcn 11640 ℕ0cn0 11900 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Σg cgsu 16716 mPoly cmpl 20135 ℂfldccnfld 20547 mDeg cmdg 24649 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-0g 16717 df-psr 20138 df-mpl 20140 df-mdeg 24651 |
This theorem is referenced by: deg1propd 24682 |
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