fprod1p - Intuitionistic Logic Explorer

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Theorem fprod1p 12110

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 10227	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
4		elfzelz 10221	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6		fzsn 10262	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
7	5, 6	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
8	7	ineq1d 3404	. . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ({𝑀} ∩ ((𝑀 + 1)...𝑁)))
9	5	zred 9569	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10	9	ltp1d 9077	. . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
11		fzdisj 10248	. . . . 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 2264	. . 3 ⊢ (𝜑 → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
14		fzsplit 10247	. . . . 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 3357	. . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
17	15, 16	eqtrd 2262	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
18		eluzelz 9731	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
19	1, 18	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20	5, 19	fzfigd 10653	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
21		elfzelz 10221	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
22		zdceq 9522	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑗 = 𝑀)
23	21, 5, 22	syl2anr 290	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 = 𝑀)
24		velsn 3683	. . . . . 6 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
25	24	dcbii 845	. . . . 5 ⊢ (DECID 𝑗 ∈ {𝑀} ↔ DECID 𝑗 = 𝑀)
26	23, 25	sylibr 134	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ {𝑀})
27	26	ralrimiva 2603	. . 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 12107	. 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 2298	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
32	28	ralrimiva 2603	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
33	31, 32, 3	rspcdva 2912	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
34	30	prodsn 12104	. . . 4 ⊢ ((𝑀 ∈ (𝑀...𝑁) ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵)
35	3, 33, 34	syl2anc 411	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵)
36	35	oveq1d 6016	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
37	29, 36	eqtrd 2262	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 839 = wceq 1395 ∈ wcel 2200 ∪ cun 3195 ∩ cin 3196 ∅c0 3491 {csn 3666 class class class wbr 4083 ‘cfv 5318 (class class class)co 6001 ℂcc 7997 1c1 8000 + caddc 8002 · cmul 8004 < clt 8181 ℤcz 9446 ℤ_≥cuz 9722 ...cfz 10204 ∏cprod 12061

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 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119

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 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-frec 6537 df-1o 6562 df-oadd 6566 df-er 6680 df-en 6888 df-dom 6889 df-fin 6890 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-fz 10205 df-fzo 10339 df-seqfrec 10670 df-exp 10761 df-ihash 10998 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-clim 11790 df-proddc 12062

This theorem is referenced by: (None)

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