mhpvarcl - Metamath Proof Explorer

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Theorem mhpvarcl 22170

Description: A power series variable is a polynomial of degree 1. (Contributed by SN, 25-May-2024.)

**Hypotheses**
Ref	Expression
mhpvarcl.h	⊢ 𝐻 = (𝐼 mHomP 𝑅)
mhpvarcl.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
mhpvarcl.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mhpvarcl.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mhpvarcl.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)

**Assertion**
Ref	Expression
mhpvarcl	⊢ (𝜑 → (𝑉‘𝑋) ∈ (𝐻‘1))

Proof of Theorem mhpvarcl Dummy variables 𝑑 ℎ 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iffalse 4540	. . . . . 6 ⊢ (¬ 𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → if(𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1_r‘𝑅), (0_g‘𝑅)) = (0_g‘𝑅))
2		mhpvarcl.v	. . . . . . . 8 ⊢ 𝑉 = (𝐼 mVar 𝑅)
3		eqid 2735	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
4		eqid 2735	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
5		eqid 2735	. . . . . . . 8 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
6		mhpvarcl.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
8		mhpvarcl.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
10		mhpvarcl.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑋 ∈ 𝐼)
12		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
13	2, 3, 4, 5, 7, 9, 11, 12	mvrval2 22021	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((𝑉‘𝑋)‘𝑑) = if(𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1_r‘𝑅), (0_g‘𝑅)))
14	13	eqeq1d 2737	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((𝑉‘𝑋)‘𝑑) = (0_g‘𝑅) ↔ if(𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1_r‘𝑅), (0_g‘𝑅)) = (0_g‘𝑅)))
15	1, 14	imbitrrid 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (¬ 𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → ((𝑉‘𝑋)‘𝑑) = (0_g‘𝑅)))
16	15	necon1ad 2955	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((𝑉‘𝑋)‘𝑑) ≠ (0_g‘𝑅) → 𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
17		nn0subm 21458	. . . . . . 7 ⊢ ℕ₀ ∈ (SubMnd‘ℂ_fld)
18		eqid 2735	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℕ₀) = (ℂ_fld ↾_s ℕ₀)
19		cnfld0 21423	. . . . . . . 8 ⊢ 0 = (0_g‘ℂ_fld)
20	18, 19	subm0 18841	. . . . . . 7 ⊢ (ℕ₀ ∈ (SubMnd‘ℂ_fld) → 0 = (0_g‘(ℂ_fld ↾_s ℕ₀)))
21	17, 20	ax-mp 5	. . . . . 6 ⊢ 0 = (0_g‘(ℂ_fld ↾_s ℕ₀))
22	18	submmnd 18839	. . . . . . 7 ⊢ (ℕ₀ ∈ (SubMnd‘ℂ_fld) → (ℂ_fld ↾_s ℕ₀) ∈ Mnd)
23	17, 22	mp1i 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (ℂ_fld ↾_s ℕ₀) ∈ Mnd)
24		eqid 2735	. . . . . 6 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))
25		1nn0 12540	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
26	18	submbas 18840	. . . . . . . . 9 ⊢ (ℕ₀ ∈ (SubMnd‘ℂ_fld) → ℕ₀ = (Base‘(ℂ_fld ↾_s ℕ₀)))
27	17, 26	ax-mp 5	. . . . . . . 8 ⊢ ℕ₀ = (Base‘(ℂ_fld ↾_s ℕ₀))
28	25, 27	eleqtri 2837	. . . . . . 7 ⊢ 1 ∈ (Base‘(ℂ_fld ↾_s ℕ₀))
29	28	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 1 ∈ (Base‘(ℂ_fld ↾_s ℕ₀)))
30	21, 23, 7, 11, 24, 29	gsummptif1n0 19999	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1)
31		oveq2 7439	. . . . . 6 ⊢ (𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
32	31	eqeq1d 2737	. . . . 5 ⊢ (𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → (((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 1 ↔ ((ℂ_fld ↾_s ℕ₀) Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1))
33	30, 32	syl5ibrcom 247	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (𝑑 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 1))
34	16, 33	syld 47	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (((𝑉‘𝑋)‘𝑑) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 1))
35	34	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} (((𝑉‘𝑋)‘𝑑) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 1))
36		mhpvarcl.h	. . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
37		eqid 2735	. . 3 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
38		eqid 2735	. . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅))
39	25	a1i 11	. . 3 ⊢ (𝜑 → 1 ∈ ℕ₀)
40	37, 2, 38, 6, 8, 10	mvrcl 22030	. . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPoly 𝑅)))
41	36, 37, 38, 4, 3, 39, 40	ismhp3 22164	. 2 ⊢ (𝜑 → ((𝑉‘𝑋) ∈ (𝐻‘1) ↔ ∀𝑑 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} (((𝑉‘𝑋)‘𝑑) ≠ (0_g‘𝑅) → ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑑) = 1)))
42	35, 41	mpbird 257	1 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (𝐻‘1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 ifcif 4531 ↦ cmpt 5231 ^◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 Fincfn 8984 0cc0 11153 1c1 11154 ℕcn 12264 ℕ₀cn0 12524 Basecbs 17245 ↾_s cress 17274 0_gc0g 17486 Σ_g cgsu 17487 Mndcmnd 18760 SubMndcsubmnd 18808 1_rcur 20199 Ringcrg 20251 ℂ_fldccnfld 21382 mVar cmvr 21943 mPoly cmpl 21944 mHomP cmhp 22151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-mgp 20153 df-ur 20200 df-ring 20253 df-cring 20254 df-cnfld 21383 df-psr 21947 df-mvr 21948 df-mpl 21949 df-mhp 22158

This theorem is referenced by: (None)

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