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Mirrors > Home > MPE Home > Th. List > vr1cl2 | Structured version Visualization version GIF version |
Description: The variable 𝑋 is a member of the power series algebra 𝑅[[𝑋]]. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
vr1val.1 | ⊢ 𝑋 = (var1‘𝑅) |
vr1cl2.2 | ⊢ 𝑆 = (PwSer1‘𝑅) |
vr1cl2.3 | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
vr1cl2 | ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vr1val.1 | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
2 | 1 | vr1val 20354 | . 2 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
3 | eqid 2821 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2821 | . . . 4 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅) | |
5 | eqid 2821 | . . . 4 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
6 | 1on 8103 | . . . . 5 ⊢ 1o ∈ On | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → 1o ∈ On) |
8 | id 22 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
9 | 0lt1o 8123 | . . . . 5 ⊢ ∅ ∈ 1o | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ ∈ 1o) |
11 | 3, 4, 5, 7, 8, 10 | mvrcl2 20200 | . . 3 ⊢ (𝑅 ∈ Ring → ((1o mVar 𝑅)‘∅) ∈ (Base‘(1o mPwSer 𝑅))) |
12 | vr1cl2.2 | . . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅) | |
13 | 12 | psr1val 20348 | . . . . 5 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
14 | 0ss 4349 | . . . . . 6 ⊢ ∅ ⊆ (1o × 1o) | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → ∅ ⊆ (1o × 1o)) |
16 | 3, 13, 15 | opsrbas 20253 | . . . 4 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘𝑆)) |
17 | vr1cl2.3 | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
18 | 16, 17 | syl6eqr 2874 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = 𝐵) |
19 | 11, 18 | eleqtrd 2915 | . 2 ⊢ (𝑅 ∈ Ring → ((1o mVar 𝑅)‘∅) ∈ 𝐵) |
20 | 2, 19 | eqeltrid 2917 | 1 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∅c0 4290 × cxp 5547 Oncon0 6185 ‘cfv 6349 (class class class)co 7150 1oc1o 8089 Basecbs 16477 Ringcrg 19291 mPwSer cmps 20125 mVar cmvr 20126 PwSer1cps1 20337 var1cv1 20338 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 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-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mgp 19234 df-ur 19246 df-ring 19293 df-psr 20130 df-mvr 20131 df-opsr 20134 df-psr1 20342 df-vr1 20343 |
This theorem is referenced by: (None) |
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