fprodm1 - Intuitionistic Logic Explorer

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Theorem fprodm1 11851

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 10225	. . . . 5 ⊢ ¬ ((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1))
2		fprodm1.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
3		eluzelz 9656	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	4	zcnd 9495	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
6		1cnd 8087	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
7	5, 6	npcand 8386	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
8	7	eleq1d 2273	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) + 1) ∈ (𝑀...(𝑁 − 1)) ↔ 𝑁 ∈ (𝑀...(𝑁 − 1))))
9	1, 8	mtbii 675	. . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ (𝑀...(𝑁 − 1)))
10		disjsn 3694	. . . 4 ⊢ (((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝑀...(𝑁 − 1)))
11	9, 10	sylibr 134	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 9652	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	2, 12	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 9409	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 9495	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17	16, 6	npcand 8386	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
18	17	fveq2d 5579	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
19	2, 18	eleqtrrd 2284	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
20		eluzp1m1 9671	. . . . . 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 10200	. . . . 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 5959	. . . 4 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
25	7	sneqd 3645	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
26	25	uneq2d 3326	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
27	23, 24, 26	3eqtr3d 2245	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
28	13, 4	fzfigd 10574	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
29		elfzelz 10146	. . . . . 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 9409	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
34	32, 33	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝑁 − 1) ∈ ℤ)
35		fzdcel 10161	. . . . 5 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → DECID 𝑗 ∈ (𝑀...(𝑁 − 1)))
36	30, 31, 34, 35	syl3anc 1249	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → DECID 𝑗 ∈ (𝑀...(𝑁 − 1)))
37	36	ralrimiva 2578	. . 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 11849	. 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 2273	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
42	38	ralrimiva 2578	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
43		eluzfz2 10153	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
44	2, 43	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
45	41, 42, 44	rspcdva 2881	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
46	40	prodsn 11846	. . . 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 5959	. 2 ⊢ (𝜑 → (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · ∏𝑘 ∈ {𝑁}𝐴) = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵))
49	39, 48	eqtrd 2237	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 835 = wceq 1372 ∈ wcel 2175 ∪ cun 3163 ∩ cin 3164 ∅c0 3459 {csn 3632 ‘cfv 5270 (class class class)co 5943 ℂcc 7922 1c1 7925 + caddc 7927 · cmul 7929 − cmin 8242 ℤcz 9371 ℤ_≥cuz 9647 ...cfz 10129 ∏cprod 11803

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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-isom 5279 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-irdg 6455 df-frec 6476 df-1o 6501 df-oadd 6505 df-er 6619 df-en 6827 df-dom 6828 df-fin 6829 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-n0 9295 df-z 9372 df-uz 9648 df-q 9740 df-rp 9775 df-fz 10130 df-fzo 10264 df-seqfrec 10591 df-exp 10682 df-ihash 10919 df-cj 11095 df-re 11096 df-im 11097 df-rsqrt 11251 df-abs 11252 df-clim 11532 df-proddc 11804

This theorem is referenced by: fprodp1 11853 fprodm1s 11854

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