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Theorem mplmon2 19314

Description: Express a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)

**Hypotheses**
Ref	Expression
mplmon2.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplmon2.v	⊢ · = ( ·_𝑠 ‘𝑃)
mplmon2.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
mplmon2.o	⊢ 1 = (1_r‘𝑅)
mplmon2.z	⊢ 0 = (0_g‘𝑅)
mplmon2.b	⊢ 𝐵 = (Base‘𝑅)
mplmon2.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplmon2.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplmon2.k	⊢ (𝜑 → 𝐾 ∈ 𝐷)
mplmon2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
mplmon2	⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))

Distinct variable groups: 𝜑,𝑦 𝑦,𝐵 𝑦,𝐷 𝑓,𝐼 𝑓,𝐾,𝑦 𝑦, 1 𝑦,𝑅 𝑦,𝑋 𝑦, 0 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐷(𝑓) 𝑃(𝑦,𝑓) 𝑅(𝑓) · (𝑦,𝑓) 1 (𝑓) 𝐼(𝑦) 𝑊(𝑦,𝑓) 𝑋(𝑓) 0 (𝑓)

**Proof of Theorem mplmon2**
Step	Hyp	Ref	Expression
1		mplmon2.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		mplmon2.v	. . 3 ⊢ · = ( ·_𝑠 ‘𝑃)
3		mplmon2.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2610	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
5		eqid 2610	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
6		mplmon2.d	. . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
7		mplmon2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		mplmon2.z	. . . 4 ⊢ 0 = (0_g‘𝑅)
9		mplmon2.o	. . . 4 ⊢ 1 = (1_r‘𝑅)
10		mplmon2.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
11		mplmon2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
12		mplmon2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
13	1, 4, 8, 9, 6, 10, 11, 12	mplmon 19284	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) ∈ (Base‘𝑃))
14	1, 2, 3, 4, 5, 6, 7, 13	mplvsca 19268	. 2 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = ((𝐷 × {𝑋}) ∘_𝑓 (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))))
15		ovex 6577	. . . . 5 ⊢ (ℕ₀ ↑_𝑚 𝐼) ∈ V
16	6, 15	rabex2 4742	. . . 4 ⊢ 𝐷 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
18	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑋 ∈ 𝐵)
19		fvex 6113	. . . . . 6 ⊢ (1_r‘𝑅) ∈ V
20	9, 19	eqeltri 2684	. . . . 5 ⊢ 1 ∈ V
21		fvex 6113	. . . . . 6 ⊢ (0_g‘𝑅) ∈ V
22	8, 21	eqeltri 2684	. . . . 5 ⊢ 0 ∈ V
23	20, 22	ifex 4106	. . . 4 ⊢ if(𝑦 = 𝐾, 1 , 0 ) ∈ V
24	23	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝐾, 1 , 0 ) ∈ V)
25		fconstmpt 5085	. . . 4 ⊢ (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋)
26	25	a1i 11	. . 3 ⊢ (𝜑 → (𝐷 × {𝑋}) = (𝑦 ∈ 𝐷 ↦ 𝑋))
27		eqidd 2611	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 )))
28	17, 18, 24, 26, 27	offval2 6812	. 2 ⊢ (𝜑 → ((𝐷 × {𝑋}) ∘_𝑓 (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))))
29		oveq2 6557	. . . . 5 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 1 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
30	29	eqeq1d 2612	. . . 4 ⊢ ( 1 = if(𝑦 = 𝐾, 1 , 0 ) → ((𝑋(._r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ) ↔ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )))
31		oveq2 6557	. . . . 5 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → (𝑋(._r‘𝑅) 0 ) = (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )))
32	31	eqeq1d 2612	. . . 4 ⊢ ( 0 = if(𝑦 = 𝐾, 1 , 0 ) → ((𝑋(._r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ) ↔ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 )))
33	3, 5, 9	ringridm 18395	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 1 ) = 𝑋)
34	11, 7, 33	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 1 ) = 𝑋)
35		iftrue 4042	. . . . . 6 ⊢ (𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 𝑋)
36	35	eqcomd 2616	. . . . 5 ⊢ (𝑦 = 𝐾 → 𝑋 = if(𝑦 = 𝐾, 𝑋, 0 ))
37	34, 36	sylan9eq 2664	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 1 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
38	3, 5, 8	ringrz 18411	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(._r‘𝑅) 0 ) = 0 )
39	11, 7, 38	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝑋(._r‘𝑅) 0 ) = 0 )
40		iffalse 4045	. . . . . 6 ⊢ (¬ 𝑦 = 𝐾 → if(𝑦 = 𝐾, 𝑋, 0 ) = 0 )
41	40	eqcomd 2616	. . . . 5 ⊢ (¬ 𝑦 = 𝐾 → 0 = if(𝑦 = 𝐾, 𝑋, 0 ))
42	39, 41	sylan9eq 2664	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑦 = 𝐾) → (𝑋(._r‘𝑅) 0 ) = if(𝑦 = 𝐾, 𝑋, 0 ))
43	30, 32, 37, 42	ifbothda 4073	. . 3 ⊢ (𝜑 → (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 )) = if(𝑦 = 𝐾, 𝑋, 0 ))
44	43	mpteq2dv 4673	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(._r‘𝑅)if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))
45	14, 28, 44	3eqtrd 2648	1 ⊢ (𝜑 → (𝑋 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝐾, 𝑋, 0 )))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ifcif 4036 {csn 4125 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ↑_𝑚 cmap 7744 Fincfn 7841 ℕcn 10897 ℕ₀cn0 11169 Basecbs 15695 ._rcmulr 15769 ·_𝑠 cvsca 15772 0_gc0g 15923 1_rcur 18324 Ringcrg 18370 mPoly cmpl 19174

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mgp 18313 df-ur 18325 df-ring 18372 df-psr 19177 df-mpl 19179

This theorem is referenced by: mplascl 19317 mplmon2cl 19321 mplmon2mul 19322 mplcoe4 19324 coe1tm 19464

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