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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 22111 | . 2 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
3 | eqid 2728 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2728 | . . . 4 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅) | |
5 | eqid 2728 | . . . 4 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
6 | 1on 8499 | . . . . 5 ⊢ 1o ∈ On | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → 1o ∈ On) |
8 | id 22 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
9 | 0lt1o 8525 | . . . . 5 ⊢ ∅ ∈ 1o | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ ∈ 1o) |
11 | 3, 4, 5, 7, 8, 10 | mvrcl2 21929 | . . 3 ⊢ (𝑅 ∈ Ring → ((1o mVar 𝑅)‘∅) ∈ (Base‘(1o mPwSer 𝑅))) |
12 | vr1cl2.2 | . . . . . 6 ⊢ 𝑆 = (PwSer1‘𝑅) | |
13 | 12 | psr1val 22105 | . . . . 5 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
14 | 0ss 4397 | . . . . . 6 ⊢ ∅ ⊆ (1o × 1o) | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → ∅ ⊆ (1o × 1o)) |
16 | 3, 13, 15 | opsrbas 21989 | . . . 4 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘𝑆)) |
17 | vr1cl2.3 | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
18 | 16, 17 | eqtr4di 2786 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = 𝐵) |
19 | 11, 18 | eleqtrd 2831 | . 2 ⊢ (𝑅 ∈ Ring → ((1o mVar 𝑅)‘∅) ∈ 𝐵) |
20 | 2, 19 | eqeltrid 2833 | 1 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ∅c0 4323 × cxp 5676 Oncon0 6369 ‘cfv 6548 (class class class)co 7420 1oc1o 8480 Basecbs 17180 Ringcrg 20173 mPwSer cmps 21837 mVar cmvr 21838 PwSer1cps1 22094 var1cv1 22095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-tset 17252 df-ple 17253 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-mgp 20075 df-ur 20122 df-ring 20175 df-psr 21842 df-mvr 21843 df-opsr 21846 df-psr1 22099 df-vr1 22100 |
This theorem is referenced by: (None) |
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