fprodunsn - Intuitionistic Logic Explorer

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Theorem fprodunsn 12030

Description: Multiply in an additional term in a finite product. See also fprodsplitsn 12059 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 3705	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
3	1, 2	sylibr 134	. . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅)
4		eqidd 2208	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵}))
5		fprodunsn.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodunsn.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		unsnfi 7042	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)
8	5, 6, 1, 7	syl3anc 1250	. . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin)
9		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴)
10	9	orcd 735	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
11		df-dc 837	. . . . . 6 ⊢ (DECID 𝑗 ∈ 𝐴 ↔ (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
12	10, 11	sylibr 134	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ 𝐴) → DECID 𝑗 ∈ 𝐴)
13		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → 𝑗 ∈ {𝐵})
14		velsn 3660	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝐵} ↔ 𝑗 = 𝐵)
15	13, 14	sylib 122	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → 𝑗 = 𝐵)
16	1	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → ¬ 𝐵 ∈ 𝐴)
17	15, 16	eqneltrd 2303	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → ¬ 𝑗 ∈ 𝐴)
18	17	olcd 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → (𝑗 ∈ 𝐴 ∨ ¬ 𝑗 ∈ 𝐴))
19	18, 11	sylibr 134	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑗 ∈ {𝐵}) → DECID 𝑗 ∈ 𝐴)
20		elun 3322	. . . . . . 7 ⊢ (𝑗 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
21	20	biimpi 120	. . . . . 6 ⊢ (𝑗 ∈ (𝐴 ∪ {𝐵}) → (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
22	21	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) → (𝑗 ∈ 𝐴 ∨ 𝑗 ∈ {𝐵}))
23	12, 19, 22	mpjaodan 800	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐴 ∪ {𝐵})) → DECID 𝑗 ∈ 𝐴)
24	23	ralrimiva 2581	. . 3 ⊢ (𝜑 → ∀𝑗 ∈ (𝐴 ∪ {𝐵})DECID 𝑗 ∈ 𝐴)
25		fprodunsn.ccl	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
26	25	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
27		elsni 3661	. . . . . . 7 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵)
28	27	adantl 277	. . . . . 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 488	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐷 ∈ ℂ)
33	30, 32	eqeltrd 2284	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ {𝐵}) → 𝐶 ∈ ℂ)
34		elun 3322	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
35	34	biimpi 120	. . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝐵}) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
36	35	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝐵}))
37	26, 33, 36	mpjaodan 800	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ)
38	3, 4, 8, 24, 37	fprodsplitdc 12022	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶))
39		fprodunsn.f	. . . . 5 ⊢ Ⅎ𝑘𝐷
40	39, 29	prodsnf 12018	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
41	6, 31, 40	syl2anc 411	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐶 = 𝐷)
42	41	oveq2d 5983	. 2 ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐶 · ∏𝑘 ∈ {𝐵}𝐶) = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))
43	38, 42	eqtrd 2240	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘 ∈ 𝐴 𝐶 · 𝐷))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 710 DECID wdc 836 = wceq 1373 ∈ wcel 2178 Ⅎwnfc 2337 ∪ cun 3172 ∩ cin 3173 ∅c0 3468 {csn 3643 (class class class)co 5967 Fincfn 6850 ℂcc 7958 · cmul 7965 ∏cprod 11976

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 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080

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 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-frec 6500 df-1o 6525 df-oadd 6529 df-er 6643 df-en 6851 df-dom 6852 df-fin 6853 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-fz 10166 df-fzo 10300 df-seqfrec 10630 df-exp 10721 df-ihash 10958 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-clim 11705 df-proddc 11977

This theorem is referenced by: fprodcl2lem 12031 fprodconst 12046 fprodap0 12047 fprodrec 12055 fprodmodd 12067

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