evl1addd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evl1addd		Structured version Visualization version GIF version

Theorem evl1addd 22328

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 2739	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
5		evl1addd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
6	2, 3, 4, 5	evl1rhm 22319	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐵)))
8		rhmghm 20455	. . . . 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 19185	. . . 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 495	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
14		evl1addd.4	. . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊))
15	14	simpld 495	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈)
16		evl1addd.u	. . . 4 ⊢ 𝑈 = (Base‘𝑃)
17		evl1addd.g	. . . 4 ⊢ ✚ = (+_g‘𝑃)
18	16, 17	grpcl 18909	. . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ✚ 𝑁) ∈ 𝑈)
19	11, 13, 15, 18	syl3anc 1379	. 2 ⊢ (𝜑 → (𝑀 ✚ 𝑁) ∈ 𝑈)
20		eqid 2739	. . . . . . 7 ⊢ (+_g‘(𝑅 ↑_s 𝐵)) = (+_g‘(𝑅 ↑_s 𝐵))
21	16, 17, 20	ghmlin 19188	. . . . . 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 1379	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)))
23		eqid 2739	. . . . . 6 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
24	5	fvexi 6842	. . . . . . 7 ⊢ 𝐵 ∈ V
25	24	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
26	16, 23	rhmf 20456	. . . . . . . 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 7027	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑_s 𝐵)))
29	27, 15	ffvelcdmd 7027	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑_s 𝐵)))
30		evl1addd.a	. . . . . 6 ⊢ + = (+_g‘𝑅)
31	4, 23, 1, 25, 28, 29, 30, 20	pwsplusgval 17446	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(+_g‘(𝑅 ↑_s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
32	22, 31	eqtrd 2774	. . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ✚ 𝑁)) = ((𝑂‘𝑀) ∘_f + (𝑂‘𝑁)))
33	32	fveq1d 6830	. . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘_f + (𝑂‘𝑁))‘𝑌))
34	4, 5, 23, 1, 25, 28	pwselbas 17444	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵)
35	34	ffnd 6657	. . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵)
36	4, 5, 23, 1, 25, 29	pwselbas 17444	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵)
37	36	ffnd 6657	. . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵)
38		evl1addd.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
39		fnfvof 7638	. . . 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 844	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘_f + (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)))
41	12	simprd 496	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉)
42	14	simprd 496	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊)
43	41, 42	oveq12d 7375	. . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) + ((𝑂‘𝑁)‘𝑌)) = (𝑉 + 𝑊))
44	33, 40, 43	3eqtrd 2778	. 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊))
45	19, 44	jca 516	1 ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ✚ 𝑁))‘𝑌) = (𝑉 + 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 Vcvv 3431 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ∘_f cof 7619 Basecbs 17171 +_gcplusg 17212 ↑_s cpws 17401 Grpcgrp 18901 GrpHom cghm 19179 CRingccrg 20207 RingHom crh 20441 Poly₁cpl1 22163 eval₁ce1 22301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-ofr 7622 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-fz 13454 df-fzo 13601 df-seq 13956 df-hash 14285 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-hom 17236 df-cco 17237 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18743 df-submnd 18744 df-grp 18904 df-minusg 18905 df-sbg 18906 df-mulg 19036 df-subg 19091 df-ghm 19180 df-cntz 19284 df-cmn 19749 df-abl 19750 df-mgp 20114 df-rng 20126 df-ur 20155 df-srg 20160 df-ring 20208 df-cring 20209 df-rhm 20444 df-subrng 20519 df-subrg 20543 df-lmod 20853 df-lss 20923 df-lsp 20963 df-assa 21829 df-asp 21830 df-ascl 21831 df-psr 21885 df-mvr 21886 df-mpl 21887 df-opsr 21889 df-evls 22051 df-evl 22052 df-psr1 22166 df-ply1 22168 df-evl1 22303

This theorem is referenced by: evl1gsumdlem 22343 evls1addd 22358 aks6d1c1p2 42603 aks6d1c1p3 42604 aks6d1c5lem1 42630 aks6d1c5lem2 42632 aks5lem3a 42683

W3C validator