Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ply1sclf1 | Structured version Visualization version GIF version |
Description: The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
ply1scl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1scl.a | ⊢ 𝐴 = (algSc‘𝑃) |
ply1sclid.k | ⊢ 𝐾 = (Base‘𝑅) |
ply1sclf1.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
ply1sclf1 | ⊢ (𝑅 ∈ Ring → 𝐴:𝐾–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1scl.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | ply1scl.a | . . 3 ⊢ 𝐴 = (algSc‘𝑃) | |
3 | ply1sclid.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
4 | ply1sclf1.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
5 | 1, 2, 3, 4 | ply1sclf 20448 | . 2 ⊢ (𝑅 ∈ Ring → 𝐴:𝐾⟶𝐵) |
6 | fveq2 6663 | . . . . 5 ⊢ ((𝐴‘𝑥) = (𝐴‘𝑦) → (coe1‘(𝐴‘𝑥)) = (coe1‘(𝐴‘𝑦))) | |
7 | 6 | fveq1d 6665 | . . . 4 ⊢ ((𝐴‘𝑥) = (𝐴‘𝑦) → ((coe1‘(𝐴‘𝑥))‘0) = ((coe1‘(𝐴‘𝑦))‘0)) |
8 | 1, 2, 3 | ply1sclid 20451 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐾) → 𝑥 = ((coe1‘(𝐴‘𝑥))‘0)) |
9 | 8 | adantrr 715 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → 𝑥 = ((coe1‘(𝐴‘𝑥))‘0)) |
10 | 1, 2, 3 | ply1sclid 20451 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐾) → 𝑦 = ((coe1‘(𝐴‘𝑦))‘0)) |
11 | 10 | adantrl 714 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → 𝑦 = ((coe1‘(𝐴‘𝑦))‘0)) |
12 | 9, 11 | eqeq12d 2836 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥 = 𝑦 ↔ ((coe1‘(𝐴‘𝑥))‘0) = ((coe1‘(𝐴‘𝑦))‘0))) |
13 | 7, 12 | syl5ibr 248 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → ((𝐴‘𝑥) = (𝐴‘𝑦) → 𝑥 = 𝑦)) |
14 | 13 | ralrimivva 3190 | . 2 ⊢ (𝑅 ∈ Ring → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 ((𝐴‘𝑥) = (𝐴‘𝑦) → 𝑥 = 𝑦)) |
15 | dff13 7006 | . 2 ⊢ (𝐴:𝐾–1-1→𝐵 ↔ (𝐴:𝐾⟶𝐵 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 ((𝐴‘𝑥) = (𝐴‘𝑦) → 𝑥 = 𝑦))) | |
16 | 5, 14, 15 | sylanbrc 585 | 1 ⊢ (𝑅 ∈ Ring → 𝐴:𝐾–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ⟶wf 6344 –1-1→wf1 6345 ‘cfv 6348 0cc0 10530 Basecbs 16478 Ringcrg 19292 algSccascl 20079 Poly1cpl1 20340 coe1cco1 20341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-ofr 7403 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-fzo 13031 df-seq 13367 df-hash 13688 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-tset 16579 df-ple 16580 df-0g 16710 df-gsum 16711 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-mulg 18220 df-subg 18271 df-ghm 18351 df-cntz 18442 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-subrg 19528 df-lmod 19631 df-lss 19699 df-ascl 20082 df-psr 20131 df-mvr 20132 df-mpl 20133 df-opsr 20135 df-psr1 20343 df-vr1 20344 df-ply1 20345 df-coe1 20346 |
This theorem is referenced by: ply1scln0 20454 mat2pmatf1 21332 facth1 24756 |
Copyright terms: Public domain | W3C validator |