fprod1p - Intuitionistic Logic Explorer

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Theorem fprod1p 12344

Description: Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)

**Hypotheses**
Ref	Expression
fprod1p.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
fprod1p.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
fprod1p.3	⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵)

**Assertion**
Ref	Expression
fprod1p	⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))

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

Proof of Theorem fprod1p Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprod1p.1	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzfz1 10414	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
4		elfzelz 10407	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6		fzsn 10450	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
7	5, 6	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
8	7	ineq1d 3431	. . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ({𝑀} ∩ ((𝑀 + 1)...𝑁)))
9	5	zred 9747	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10	9	ltp1d 9250	. . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
11		fzdisj 10435	. . . . 5 ⊢ (𝑀 < (𝑀 + 1) → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
12	10, 11	syl 14	. . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
13	8, 12	eqtr3d 2273	. . 3 ⊢ (𝜑 → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
14		fzsplit 10434	. . . . 5 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)))
15	3, 14	syl 14	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)))
16	7	uneq1d 3382	. . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
17	15, 16	eqtrd 2271	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
18		eluzelz 9910	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
19	1, 18	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20	5, 19	fzfigd 10846	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
21		elfzelz 10407	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
22		zdceq 9699	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑗 = 𝑀)
23	21, 5, 22	syl2anr 290	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 = 𝑀)
24		velsn 3722	. . . . . 6 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
25	24	dcbii 852	. . . . 5 ⊢ (DECID 𝑗 ∈ {𝑀} ↔ DECID 𝑗 = 𝑀)
26	23, 25	sylibr 134	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ {𝑀})
27	26	ralrimiva 2623	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝑁)DECID 𝑗 ∈ {𝑀})
28		fprod1p.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
29	13, 17, 20, 27, 28	fprodsplitdc 12341	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
30		fprod1p.3	. . . . . 6 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵)
31	30	eleq1d 2307	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
32	28	ralrimiva 2623	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
33	31, 32, 3	rspcdva 2934	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
34	30	prodsn 12338	. . . 4 ⊢ ((𝑀 ∈ (𝑀...𝑁) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵)
35	3, 33, 34	syl2anc 415	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵)
36	35	oveq1d 6090	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
37	29, 36	eqtrd 2271	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ∪ cun 3218 ∩ cin 3219 ∅c0 3520 {csn 3705 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 ℂcc 8167 1c1 8170 + caddc 8172 · cmul 8174 < clt 8350 ℤcz 9623 ℤ_≥cuz 9900 ...cfz 10390 ∏cprod 12295

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-oadd 6681 df-er 6797 df-en 7013 df-dom 7014 df-fin 7015 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-fz 10391 df-fzo 10528 df-seqfrec 10863 df-exp 10954 df-ihash 11193 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-proddc 12296

This theorem is referenced by: (None)

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