evl1addd - Metamath Proof Explorer

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Theorem evl1addd 22302

Description: Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)

**Hypotheses**
Ref	Expression
evl1addd.q	⊢ 𝑂 = (eval₁‘𝑅)
evl1addd.p	⊢ 𝑃 = (Poly₁‘𝑅)
evl1addd.b	⊢ 𝐵 = (Base‘𝑅)
evl1addd.u	⊢ 𝑈 = (Base‘𝑃)
evl1addd.1	⊢ (𝜑 → 𝑅 ∈ CRing)
evl1addd.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)
evl1addd.3	⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))
evl1addd.4	⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))
evl1addd.g	⊢ ✚ = (+_g‘𝑃)
evl1addd.a	⊢ + = (+_g‘𝑅)

**Assertion**
Ref	Expression
evl1addd	⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)))

**Proof of Theorem evl1addd**
Step	Hyp	Ref	Expression
1		evl1addd.1	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
2		evl1addd.q	. . . . . . 7 ⊢ 𝑂 = (eval₁‘𝑅)
3		evl1addd.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
4		eqid 2737	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
5		evl1addd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
6	2, 3, 4, 5	evl1rhm 22293	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
8		rhmghm 20436	. . . . 5 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)) → 𝑂 ∈ (𝑃 GrpHom (𝑅 ↑_s 𝐵)))
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 GrpHom (𝑅 ↑_s 𝐵)))
10		ghmgrp1 19164	. . . 4 ⊢ (𝑂 ∈ (𝑃 GrpHom (𝑅 ↑_s 𝐵)) → 𝑃 ∈ Grp)
11	9, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ Grp)
12		evl1addd.3	. . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉))
13	12	simpld 494	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
14		evl1addd.4	. . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))
15	14	simpld 494	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈)
16		evl1addd.u	. . . 4 ⊢ 𝑈 = (Base‘𝑃)
17		evl1addd.g	. . . 4 ⊢ ✚ = (+_g‘𝑃)
18	16, 17	grpcl 18888	. . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ✚ 𝑁) ∈ 𝑈)
19	11, 13, 15, 18	syl3anc 1374	. 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈)
20		eqid 2737	. . . . . . 7 ⊢ (+_g‘(𝑅 ↑_s 𝐵)) = (+_g‘(𝑅 ↑_s 𝐵))
21	16, 17, 20	ghmlin 19167	. . . . . 6 ⊢ ((𝑂 ∈ (𝑃 GrpHom (𝑅 ↑_s 𝐵)) ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)))
22	9, 13, 15, 21	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)))
23		eqid 2737	. . . . . 6 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
24	5	fvexi 6858	. . . . . . 7 ⊢ 𝐵 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
26	16, 23	rhmf 20437	. . . . . . . 8 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑_s 𝐵)))
27	7, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑_s 𝐵)))
28	27, 13	ffvelcdmd 7041	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑_s 𝐵)))
29	27, 15	ffvelcdmd 7041	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑_s 𝐵)))
30		evl1addd.a	. . . . . 6 ⊢ + = (+_g‘𝑅)
31	4, 23, 1, 25, 28, 29, 30, 20	pwsplusgval 17425	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
32	22, 31	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
33	32	fveq1d 6846	. . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘_f + (𝑂‘𝑁))‘𝑌))
34	4, 5, 23, 1, 25, 28	pwselbas 17423	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵)
35	34	ffnd 6673	. . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵)
36	4, 5, 23, 1, 25, 29	pwselbas 17423	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵)
37	36	ffnd 6673	. . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵)
38		evl1addd.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
39		fnfvof 7651	. . . 4 ⊢ ((((𝑂‘𝑀) Fn 𝐵 ∧ (𝑂‘𝑁) Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝑌 ∈ 𝐵)) → (((𝑂‘𝑀) ∘_f + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)))
40	35, 37, 25, 38, 39	syl22anc 839	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘_f + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)))
41	12	simprd 495	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉)
42	14	simprd 495	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊)
43	41, 42	oveq12d 7388	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)) = (𝑉 + 𝑊))
44	33, 40, 43	3eqtrd 2776	. 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊))
45	19, 44	jca 511	1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 Fn wfn 6497 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ∘_f cof 7632 Basecbs 17150 +_gcplusg 17191 ↑_s cpws 17380 Grpcgrp 18880 GrpHom cghm 19158 CRingccrg 20186 RingHom crh 20422 Poly₁cpl1 22134 eval₁ce1 22275

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-ofr 7635 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-sup 9359 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-fz 13438 df-fzo 13585 df-seq 13939 df-hash 14268 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-hom 17215 df-cco 17216 df-0g 17375 df-gsum 17376 df-prds 17381 df-pws 17383 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-mhm 18722 df-submnd 18723 df-grp 18883 df-minusg 18884 df-sbg 18885 df-mulg 19015 df-subg 19070 df-ghm 19159 df-cntz 19263 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-srg 20139 df-ring 20187 df-cring 20188 df-rhm 20425 df-subrng 20496 df-subrg 20520 df-lmod 20830 df-lss 20900 df-lsp 20940 df-assa 21825 df-asp 21826 df-ascl 21827 df-psr 21882 df-mvr 21883 df-mpl 21884 df-opsr 21886 df-evls 22046 df-evl 22047 df-psr1 22137 df-ply1 22139 df-evl1 22277

This theorem is referenced by: evl1gsumdlem 22317 evls1addd 22332 aks6d1c1p2 42508 aks6d1c1p3 42509 aks6d1c5lem1 42535 aks6d1c5lem2 42537 aks5lem3a 42588

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