fprodp1s - Metamath Proof Explorer

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Theorem fprodp1s 15973

Description: Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)

**Hypotheses**
Ref	Expression
fprodp1s.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
fprodp1s.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)

**Assertion**
Ref	Expression
fprodp1s	⊢ (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ⦋(𝑁 + 1) / 𝑘⦌𝐴))

Distinct variable groups: 𝜑,𝑘 𝑘,𝑀 𝑘,𝑁 Allowed substitution hint: 𝐴(𝑘)

Proof of Theorem fprodp1s Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodp1s.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		fprodp1s.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)
3	2	ralrimiva 3136	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 ∈ ℂ)
4		nfcsb1v 3917	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴
5	4	nfel1 2909	. . . . 5 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ
6		csbeq1a 3906	. . . . . 6 ⊢ (𝑘 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑘⦌𝐴)
7	6	eleq1d 2811	. . . . 5 ⊢ (𝑘 = 𝑚 → (𝐴 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ))
8	5, 7	rspc 3596	. . . 4 ⊢ (𝑚 ∈ (𝑀...(𝑁 + 1)) → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 ∈ ℂ → ⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ))
9	3, 8	mpan9 505	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...(𝑁 + 1))) → ⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ)
10		csbeq1 3895	. . 3 ⊢ (𝑚 = (𝑁 + 1) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋(𝑁 + 1) / 𝑘⦌𝐴)
11	1, 9, 10	fprodp1 15971	. 2 ⊢ (𝜑 → ∏𝑚 ∈ (𝑀...(𝑁 + 1))⦋𝑚 / 𝑘⦌𝐴 = (∏𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑘⦌𝐴 · ⦋(𝑁 + 1) / 𝑘⦌𝐴))
12		nfcv 2892	. . 3 ⊢ Ⅎ𝑚𝐴
13	12, 4, 6	cbvprodi 15919	. 2 ⊢ ∏𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = ∏𝑚 ∈ (𝑀...(𝑁 + 1))⦋𝑚 / 𝑘⦌𝐴
14	12, 4, 6	cbvprodi 15919	. . 3 ⊢ ∏𝑘 ∈ (𝑀...𝑁)𝐴 = ∏𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑘⦌𝐴
15	14	oveq1i 7434	. 2 ⊢ (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ⦋(𝑁 + 1) / 𝑘⦌𝐴) = (∏𝑚 ∈ (𝑀...𝑁)⦋𝑚 / 𝑘⦌𝐴 · ⦋(𝑁 + 1) / 𝑘⦌𝐴)
16	11, 13, 15	3eqtr4g 2791	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ⦋(𝑁 + 1) / 𝑘⦌𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⦋csb 3892 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 1c1 11159 + caddc 11161 · cmul 11163 ℤ_≥cuz 12874 ...cfz 13538 ∏cprod 15907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-prod 15908

This theorem is referenced by: fprodabs 15976

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