fprodm1 - Intuitionistic Logic Explorer

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Theorem fprodm1 11488

Description: Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)

**Hypotheses**
Ref	Expression
fprodm1.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
fprodm1.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
fprodm1.3	⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵)

**Assertion**
Ref	Expression
fprodm1	⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵))

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

Proof of Theorem fprodm1 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzp1nel 9999	. . . . 5 ⊢ ¬ ((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1))
2		fprodm1.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
3		eluzelz 9442	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	4	zcnd 9281	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
6		1cnd 7888	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
7	5, 6	npcand 8184	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
8	7	eleq1d 2226	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) ↔ 𝑁 ∈ (𝑀...(𝑁 − 1))))
9	1, 8	mtbii 664	. . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1)))
10		disjsn 3621	. . . 4 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝑀...(𝑁 − 1)))
11	9, 10	sylibr 133	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 9438	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	2, 12	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 9199	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 9281	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17	16, 6	npcand 8184	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
18	17	fveq2d 5471	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
19	2, 18	eleqtrrd 2237	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
20		eluzp1m1 9456	. . . . . 6 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
21	15, 19, 20	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
22		fzsuc2 9974	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
23	13, 21, 22	syl2anc 409	. . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
24	7	oveq2d 5837	. . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
25	7	sneqd 3573	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
26	25	uneq2d 3261	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
27	23, 24, 26	3eqtr3d 2198	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
28	13, 4	fzfigd 10323	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
29		elfzelz 9921	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
30	29	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℤ)
31	13	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
32	4	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ)
33		peano2zm 9199	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
34	32, 33	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝑁 − 1) ∈ ℤ)
35		fzdcel 9935	. . . . 5 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → DECID 𝑗 ∈ (𝑀...(𝑁 − 1)))
36	30, 31, 34, 35	syl3anc 1220	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ (𝑀...(𝑁 − 1)))
37	36	ralrimiva 2530	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)DECID 𝑗 ∈ (𝑀...(𝑁 − 1)))
38		fprodm1.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
39	11, 27, 28, 37, 38	fprodsplitdc 11486	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴))
40		fprodm1.3	. . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵)
41	40	eleq1d 2226	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
42	38	ralrimiva 2530	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
43		eluzfz2 9927	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
44	2, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
45	41, 42, 44	rspcdva 2821	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
46	40	prodsn 11483	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵)
47	2, 45, 46	syl2anc 409	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵)
48	47	oveq2d 5837	. 2 ⊢ (𝜑 → (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴) = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵))
49	39, 48	eqtrd 2190	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ∪ cun 3100 ∩ cin 3101 ∅c0 3394 {csn 3560 ‘cfv 5169 (class class class)co 5821 ℂcc 7724 1c1 7727 + caddc 7729 · cmul 7731 − cmin 8040 ℤcz 9161 ℤ_≥cuz 9433 ...cfz 9905 ∏cprod 11440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843 ax-pre-mulext 7844 ax-arch 7845 ax-caucvg 7846

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-isom 5178 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-frec 6335 df-1o 6360 df-oadd 6364 df-er 6477 df-en 6683 df-dom 6684 df-fin 6685 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-reap 8444 df-ap 8451 df-div 8540 df-inn 8828 df-2 8886 df-3 8887 df-4 8888 df-n0 9085 df-z 9162 df-uz 9434 df-q 9522 df-rp 9554 df-fz 9906 df-fzo 10035 df-seqfrec 10338 df-exp 10412 df-ihash 10643 df-cj 10735 df-re 10736 df-im 10737 df-rsqrt 10891 df-abs 10892 df-clim 11169 df-proddc 11441

This theorem is referenced by: fprodp1 11490 fprodm1s 11491

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