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Mirrors > Home > MPE Home > Th. List > psr1vsca | Structured version Visualization version GIF version |
Description: Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
psr1plusg.y | ⊢ 𝑌 = (PwSer1‘𝑅) |
psr1plusg.s | ⊢ 𝑆 = (1o mPwSer 𝑅) |
psr1vscafval.n | ⊢ · = ( ·𝑠 ‘𝑌) |
Ref | Expression |
---|---|
psr1vsca | ⊢ · = ( ·𝑠 ‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psr1vscafval.n | . 2 ⊢ · = ( ·𝑠 ‘𝑌) | |
2 | psr1plusg.s | . . . 4 ⊢ 𝑆 = (1o mPwSer 𝑅) | |
3 | psr1plusg.y | . . . . 5 ⊢ 𝑌 = (PwSer1‘𝑅) | |
4 | 3 | psr1val 20348 | . . . 4 ⊢ 𝑌 = ((1o ordPwSer 𝑅)‘∅) |
5 | 0ss 4350 | . . . . 5 ⊢ ∅ ⊆ (1o × 1o) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ⊆ (1o × 1o)) |
7 | 2, 4, 6 | opsrvsca 20256 | . . 3 ⊢ (⊤ → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑌)) |
8 | 7 | mptru 1540 | . 2 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑌) |
9 | 1, 8 | eqtr4i 2847 | 1 ⊢ · = ( ·𝑠 ‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ⊆ wss 3936 ∅c0 4291 × cxp 5548 ‘cfv 6350 (class class class)co 7150 1oc1o 8089 ·𝑠 cvsca 16563 mPwSer cmps 20125 PwSer1cps1 20337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 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-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-vsca 16576 df-ple 16579 df-psr 20130 df-opsr 20134 df-psr1 20342 |
This theorem is referenced by: ply1vsca 20388 |
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