psrcrng - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > psrcrng		Structured version Visualization version GIF version

Theorem psrcrng 22312

Description: The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
psrcnrg.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrcnrg.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrcnrg.r	⊢ (𝜑 → 𝑅 ∈ CRing)

**Assertion**
Ref	Expression
psrcrng	⊢ (𝜑 → 𝑆 ∈ CRing)

Proof of Theorem psrcrng Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrcnrg.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		psrcnrg.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
3		psrcnrg.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing)
4		crngring 20505	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	1, 2, 5	psrring 22310	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
7		eqid 2763	. . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
8		eqid 2763	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	7, 8	mgpbas 20398	. . . 4 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆))
10	9	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(mulGrp‘𝑆)))
11		eqid 2763	. . . . 5 ⊢ (._r‘𝑆) = (._r‘𝑆)
12	7, 11	mgpplusg 20397	. . . 4 ⊢ (._r‘𝑆) = (+_g‘(mulGrp‘𝑆))
13	12	a1i 11	. . 3 ⊢ (𝜑 → (._r‘𝑆) = (+_g‘(mulGrp‘𝑆)))
14	7	ringmgp 20498	. . . 4 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
15	6, 14	syl 18	. . 3 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
16	2	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉)
17	5	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
18		eqid 2763	. . . 4 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
19		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20		simp3 1156	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ CRing)
22	1, 16, 17, 18, 11, 8, 19, 20, 21	psrcom 22308	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(._r‘𝑆)𝑦) = (𝑦(._r‘𝑆)𝑥))
23	10, 13, 15, 22	iscmnd 20041	. 2 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd)
24	7	iscrng 20499	. 2 ⊢ (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ (mulGrp‘𝑆) ∈ CMnd))
25	6, 23, 24	sylanbrc 595	1 ⊢ (𝜑 → 𝑆 ∈ CRing)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {crab 3416 ^◡ccnv 5655 “ cima 5659 ‘cfv 6561 (class class class)co 7446 ↑_m cmap 8878 Fincfn 9004 ℕcn 12365 ℕ₀cn0 12636 Basecbs 17419 +_gcplusg 17460 ._rcmulr 17461 Mndcmnd 18955 CMndccmn 20027 mulGrpcmgp 20393 Ringcrg 20492 CRingccrg 20493 mPwSer cmps 22245

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-ofr 7721 df-om 7905 df-1st 8028 df-2nd 8029 df-supp 8208 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-map 8880 df-pm 8881 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-fsupp 9385 df-sup 9464 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-fz 13670 df-fzo 13819 df-seq 14175 df-hash 14505 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-sca 17476 df-vsca 17477 df-ip 17478 df-tset 17479 df-ple 17480 df-ds 17482 df-hom 17484 df-cco 17485 df-0g 17644 df-gsum 17645 df-prds 17650 df-pws 17652 df-mre 17788 df-mrc 17789 df-acs 17791 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-mhm 19010 df-submnd 19011 df-grp 19179 df-minusg 19180 df-mulg 19310 df-ghm 19460 df-cntz 19564 df-cmn 20029 df-abl 20030 df-mgp 20394 df-rng 20408 df-ur 20441 df-ring 20494 df-cring 20495 df-psr 22250

This theorem is used by: mplcrng 22361 opsrcrng 22401 psd1 22521 psdpw 22524 psrmonprod 34267

W3C validator