fprodunsn - Intuitionistic Logic Explorer

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Theorem fprodunsn 11545

Description: Multiply in an additional term in a finite product. See also fprodsplitsn 11574 which is the same but with a Ⅎ𝑘𝜑 hypothesis in place of the distinct variable condition between 𝜑 and 𝑘. (Contributed by Jim Kingdon, 16-Aug-2024.)

**Hypotheses**
Ref	Expression
fprodunsn.f	⊢ Ⅎ𝑘𝐷
fprodunsn.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodunsn.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fprodunsn.ba	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)
fprodunsn.ccl	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
fprodunsn.dcl	⊢ (𝜑 → 𝐷 ∈ ℂ)
fprodunsn.d	⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)

**Assertion**
Ref	Expression
fprodunsn	⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑉 𝜑,𝑘 Allowed substitution hints: 𝐶(𝑘) 𝐷(𝑘)

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

Step	Hyp	Ref	Expression
1		fprodunsn.ba	. . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)
2		disjsn 3638	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
3	1, 2	sylibr 133	. . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅)
4		eqidd 2166	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵}))
5		fprodunsn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodunsn.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		unsnfi 6884	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)
8	5, 6, 1, 7	syl3anc 1228	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin)
9		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴)
10	9	orcd 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
11		df-dc 825	. . . . . 6 ⊢ (DECID 𝑗 ∈ 𝐴 ↔ (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
12	10, 11	sylibr 133	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → DECID 𝑗 ∈ 𝐴)
13		simpr 109	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → 𝑗 ∈ {𝐵})
14		velsn 3593	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝐵} ↔ 𝑗 = 𝐵)
15	13, 14	sylib 121	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → 𝑗 = 𝐵)
16	1	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → ¬ 𝐵 ∈ 𝐴)
17	15, 16	eqneltrd 2262	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → ¬ 𝑗 ∈ 𝐴)
18	17	olcd 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
19	18, 11	sylibr 133	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → DECID 𝑗 ∈ 𝐴)
20		elun 3263	. . . . . . 7 ⊢ (𝑗 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
21	20	biimpi 119	. . . . . 6 ⊢ (𝑗 ∈ (𝐴 ∪ {𝐵}) → (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
22	21	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) → (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
23	12, 19, 22	mpjaodan 788	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) → DECID 𝑗 ∈ 𝐴)
24	23	ralrimiva 2539	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ (𝐴 ∪ {𝐵})DECID 𝑗 ∈ 𝐴)
25		fprodunsn.ccl	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
26	25	adantlr 469	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
27		elsni 3594	. . . . . . 7 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵)
28	27	adantl 275	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝑘 = 𝐵)
29		fprodunsn.d	. . . . . 6 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)
30	28, 29	syl 14	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐶 = 𝐷)
31		fprodunsn.dcl	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ)
32	31	ad2antrr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐷 ∈ ℂ)
33	30, 32	eqeltrd 2243	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐶 ∈ ℂ)
34		elun 3263	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
35	34	biimpi 119	. . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
36	35	adantl 275	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
37	26, 33, 36	mpjaodan 788	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ)
38	3, 4, 8, 24, 37	fprodsplitdc 11537	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶))
39		fprodunsn.f	. . . . 5 ⊢ Ⅎ𝑘𝐷
40	39, 29	prodsnf 11533	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
41	6, 31, 40	syl2anc 409	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
42	41	oveq2d 5858	. 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
43	38, 42	eqtrd 2198	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 824 = wceq 1343 ∈ wcel 2136 Ⅎwnfc 2295 ∪ cun 3114 ∩ cin 3115 ∅c0 3409 {csn 3576 (class class class)co 5842 Fincfn 6706 ℂcc 7751 · cmul 7758 ∏cprod 11491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-en 6707 df-dom 6708 df-fin 6709 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-ihash 10689 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-proddc 11492

This theorem is referenced by: fprodcl2lem 11546 fprodconst 11561 fprodap0 11562 fprodrec 11570 fprodmodd 11582

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