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Mirrors > Home > MPE Home > Th. List > deg1fval | Structured version Visualization version GIF version |
Description: Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
deg1fval.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
Ref | Expression |
---|---|
deg1fval | ⊢ 𝐷 = (1o mDeg 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1fval.d | . 2 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | oveq2 7369 | . . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅)) | |
3 | df-deg1 25441 | . . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟)) | |
4 | ovex 7394 | . . . 4 ⊢ (1o mDeg 𝑅) ∈ V | |
5 | 2, 3, 4 | fvmpt 6952 | . . 3 ⊢ (𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅)) |
6 | fvprc 6838 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = ∅) | |
7 | reldmmdeg 25442 | . . . . 5 ⊢ Rel dom mDeg | |
8 | 7 | ovprc2 7401 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅) |
9 | 6, 8 | eqtr4d 2776 | . . 3 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅)) |
10 | 5, 9 | pm2.61i 182 | . 2 ⊢ ( deg1 ‘𝑅) = (1o mDeg 𝑅) |
11 | 1, 10 | eqtri 2761 | 1 ⊢ 𝐷 = (1o mDeg 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 ‘cfv 6500 (class class class)co 7361 1oc1o 8409 mDeg cmdg 25438 deg1 cdg1 25439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-mdeg 25440 df-deg1 25441 |
This theorem is referenced by: deg1xrf 25469 deg1cl 25471 deg1propd 25474 deg1z 25475 deg1nn0cl 25476 deg1ldg 25480 deg1leb 25483 deg1val 25484 deg1addle 25489 deg1vscale 25492 deg1vsca 25493 deg1mulle2 25497 deg1le0 25499 |
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