fprodm1 - Intuitionistic Logic Explorer

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Theorem fprodm1 12104

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 10296	. . . . 5 ⊢ ¬ ((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1))
2		fprodm1.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
3		eluzelz 9727	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	4	zcnd 9566	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
6		1cnd 8158	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
7	5, 6	npcand 8457	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
8	7	eleq1d 2298	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) ↔ 𝑁 ∈ (𝑀...(𝑁 − 1))))
9	1, 8	mtbii 678	. . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1)))
10		disjsn 3728	. . . 4 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝑀...(𝑁 − 1)))
11	9, 10	sylibr 134	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 9723	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	2, 12	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 9480	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 9566	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17	16, 6	npcand 8457	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
18	17	fveq2d 5630	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
19	2, 18	eleqtrrd 2309	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
20		eluzp1m1 9742	. . . . . 6 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
21	15, 19, 20	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
22		fzsuc2 10271	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
23	13, 21, 22	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
24	7	oveq2d 6016	. . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
25	7	sneqd 3679	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
26	25	uneq2d 3358	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
27	23, 24, 26	3eqtr3d 2270	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
28	13, 4	fzfigd 10648	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
29		elfzelz 10217	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
30	29	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℤ)
31	13	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
32	4	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ)
33		peano2zm 9480	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
34	32, 33	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝑁 − 1) ∈ ℤ)
35		fzdcel 10232	. . . . 5 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → DECID 𝑗 ∈ (𝑀...(𝑁 − 1)))
36	30, 31, 34, 35	syl3anc 1271	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ (𝑀...(𝑁 − 1)))
37	36	ralrimiva 2603	. . 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 12102	. 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 2298	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
42	38	ralrimiva 2603	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
43		eluzfz2 10224	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
44	2, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
45	41, 42, 44	rspcdva 2912	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
46	40	prodsn 12099	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵)
47	2, 45, 46	syl2anc 411	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑁}𝐴 = 𝐵)
48	47	oveq2d 6016	. 2 ⊢ (𝜑 → (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴) = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵))
49	39, 48	eqtrd 2262	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 104 DECID wdc 839 = wceq 1395 ∈ wcel 2200 ∪ cun 3195 ∩ cin 3196 ∅c0 3491 {csn 3666 ‘cfv 5317 (class class class)co 6000 ℂcc 7993 1c1 7996 + caddc 7998 · cmul 8000 − cmin 8313 ℤcz 9442 ℤ_≥cuz 9718 ...cfz 10200 ∏cprod 12056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-frec 6535 df-1o 6560 df-oadd 6564 df-er 6678 df-en 6886 df-dom 6887 df-fin 6888 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-fz 10201 df-fzo 10335 df-seqfrec 10665 df-exp 10756 df-ihash 10993 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-clim 11785 df-proddc 12057

This theorem is referenced by: fprodp1 12106 fprodm1s 12107

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