fprod1p - Intuitionistic Logic Explorer

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Theorem fprod1p 11943

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 10155	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
4		elfzelz 10149	. . . . . . 7 ⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6		fzsn 10190	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
7	5, 6	syl 14	. . . . 5 ⊢ (𝜑 → (𝑀...𝑀) = {𝑀})
8	7	ineq1d 3373	. . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ({𝑀} ∩ ((𝑀 + 1)...𝑁)))
9	5	zred 9497	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10	9	ltp1d 9005	. . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
11		fzdisj 10176	. . . . 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 2240	. . 3 ⊢ (𝜑 → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
14		fzsplit 10175	. . . . 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 3326	. . . 4 ⊢ (𝜑 → ((𝑀...𝑀) ∪ ((𝑀 + 1)...𝑁)) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
17	15, 16	eqtrd 2238	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
18		eluzelz 9659	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
19	1, 18	syl 14	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20	5, 19	fzfigd 10578	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
21		elfzelz 10149	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
22		zdceq 9450	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑗 = 𝑀)
23	21, 5, 22	syl2anr 290	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 = 𝑀)
24		velsn 3650	. . . . . 6 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
25	24	dcbii 842	. . . . 5 ⊢ (DECID 𝑗 ∈ {𝑀} ↔ DECID 𝑗 = 𝑀)
26	23, 25	sylibr 134	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ {𝑀})
27	26	ralrimiva 2579	. . 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 11940	. 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 2274	. . . . 5 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
32	28	ralrimiva 2579	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
33	31, 32, 3	rspcdva 2882	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
34	30	prodsn 11937	. . . 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 5961	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝑀}𝐴 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
37	29, 36	eqtrd 2238	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 836 = wceq 1373 ∈ wcel 2176 ∪ cun 3164 ∩ cin 3165 ∅c0 3460 {csn 3633 class class class wbr 4045 ‘cfv 5272 (class class class)co 5946 ℂcc 7925 1c1 7928 + caddc 7930 · cmul 7932 < clt 8109 ℤcz 9374 ℤ_≥cuz 9650 ...cfz 10132 ∏cprod 11894

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045 ax-arch 8046 ax-caucvg 8047

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-isom 5281 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-1st 6228 df-2nd 6229 df-recs 6393 df-irdg 6458 df-frec 6479 df-1o 6504 df-oadd 6508 df-er 6622 df-en 6830 df-dom 6831 df-fin 6832 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-2 9097 df-3 9098 df-4 9099 df-n0 9298 df-z 9375 df-uz 9651 df-q 9743 df-rp 9778 df-fz 10133 df-fzo 10267 df-seqfrec 10595 df-exp 10686 df-ihash 10923 df-cj 11186 df-re 11187 df-im 11188 df-rsqrt 11342 df-abs 11343 df-clim 11623 df-proddc 11895

This theorem is referenced by: (None)

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