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Mirrors > Home > MPE Home > Th. List > cpmatelimp | Structured version Visualization version Unicode version |
Description: Implication of a set being a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.) |
Ref | Expression |
---|---|
cpmat.s | ConstPolyMat |
cpmat.p | Poly1 |
cpmat.c | Mat |
cpmat.b |
Ref | Expression |
---|---|
cpmatelimp | coe1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmat.s | . . . . 5 ConstPolyMat | |
2 | cpmat.p | . . . . 5 Poly1 | |
3 | cpmat.c | . . . . 5 Mat | |
4 | cpmat.b | . . . . 5 | |
5 | 1, 2, 3, 4 | cpmatpmat 20515 | . . . 4 |
6 | 5 | 3expa 1265 | . . 3 |
7 | 1, 2, 3, 4 | cpmatel 20516 | . . . . . 6 coe1 |
8 | 7 | 3expa 1265 | . . . . 5 coe1 |
9 | 8 | biimpd 219 | . . . 4 coe1 |
10 | 9 | impancom 456 | . . 3 coe1 |
11 | 6, 10 | jcai 559 | . 2 coe1 |
12 | 11 | ex 450 | 1 coe1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cfv 5888 (class class class)co 6650 cfn 7955 cn 11020 cbs 15857 c0g 16100 crg 18547 Poly1cpl1 19547 coe1cco1 19548 Mat cmat 20213 ConstPolyMat ccpmat 20508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-cpmat 20511 |
This theorem is referenced by: cpmatmcllem 20523 m2cpminvid2lem 20559 |
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