Theorem remulscllem1 28403

Description: Lemma for remulscl 28405. Split a product of reciprocals of naturals. (Contributed by Scott Fenton, 16-Apr-2025.)

Assertion

Ref	Expression
remulscllem1	⊢ (∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)))

Distinct variable groups:   𝐴,𝑝,𝑞,𝑛   𝐵,𝑝,𝑞,𝑛   𝐹,𝑝,𝑞,𝑛

Proof of Theorem remulscllem1

Step	Hyp	Ref	Expression
1		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = (𝑝 ·s 𝑞) → ( 1s /su 𝑛) = ( 1s /su (𝑝 ·s 𝑞)))
2	1	oveq2d 7421	. . . . . 6 ⊢ (𝑛 = (𝑝 ·s 𝑞) → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
3	2	eqeq2d 2746	. . . . 5 ⊢ (𝑛 = (𝑝 ·s 𝑞) → ((𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛)) ↔ (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞)))))
4		nnmulscl 28291	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝑝 ·s 𝑞) ∈ ℕs)
5		1sno 27791	. . . . . . . . 9 ⊢ 1s ∈ No
6	5	a1i 11	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 1s ∈ No )
7		nnsno 28269	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ∈ No )
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ∈ No )
9		nnsno 28269	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ∈ No )
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ∈ No )
11		nnne0s 28281	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ≠ 0s )
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ≠ 0s )
13		nnne0s 28281	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ≠ 0s )
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ≠ 0s )
15	6, 8, 6, 10, 12, 14	divmuldivsd 28186	. . . . . . 7 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)))
16		mulsrid 28068	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
17	5, 16	ax-mp 5	. . . . . . . 8 ⊢ ( 1s ·s 1s ) = 1s
18	17	oveq1i 7415	. . . . . . 7 ⊢ (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)) = ( 1s /su (𝑝 ·s 𝑞))
19	15, 18	eqtrdi 2786	. . . . . 6 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = ( 1s /su (𝑝 ·s 𝑞)))
20	19	oveq2d 7421	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
21	3, 4, 20	rspcedvdw 3604	. . . 4 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → ∃𝑛 ∈ ℕs (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛)))
22		eqeq1 2739	. . . . 5 ⊢ (𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → (𝐴 = (𝐵𝐹( 1s /su 𝑛)) ↔ (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛))))
23	22	rexbidv 3164	. . . 4 ⊢ (𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → (∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛))))
24	21, 23	syl5ibrcom 247	. . 3 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛))))
25	24	rexlimivv 3186	. 2 ⊢ (∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)))
26	5	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
27		nnsno 28269	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
28		nnne0s 28281	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
29	26, 27, 28	divscld 28178	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
30	29	mulsridd 28069	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (( 1s /su 𝑛) ·s 1s ) = ( 1s /su 𝑛))
31	30	eqcomd 2741	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) = (( 1s /su 𝑛) ·s 1s ))
32	31	oveq2d 7421	. . . . 5 ⊢ (𝑛 ∈ ℕs → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
33		1nns 28293	. . . . . 6 ⊢ 1s ∈ ℕs
34		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
35	34	oveq1d 7420	. . . . . . . . 9 ⊢ (𝑝 = 𝑛 → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑞)))
36	35	oveq2d 7421	. . . . . . . 8 ⊢ (𝑝 = 𝑛 → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))))
37	36	eqeq2d 2746	. . . . . . 7 ⊢ (𝑝 = 𝑛 → ((𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞)))))
38		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = ( 1s /su 1s ))
39		divs1 28159	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
40	5, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s /su 1s ) = 1s
41	38, 40	eqtrdi 2786	. . . . . . . . . 10 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = 1s )
42	41	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑞 = 1s → (( 1s /su 𝑛) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s 1s ))
43	42	oveq2d 7421	. . . . . . . 8 ⊢ (𝑞 = 1s → (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
44	43	eqeq2d 2746	. . . . . . 7 ⊢ (𝑞 = 1s → ((𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))) ↔ (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s ))))
45	37, 44	rspc2ev 3614	. . . . . 6 ⊢ ((𝑛 ∈ ℕs ∧ 1s ∈ ℕs ∧ (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s ))) → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
46	33, 45	mp3an2 1451	. . . . 5 ⊢ ((𝑛 ∈ ℕs ∧ (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s ))) → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
47	32, 46	mpdan 687	. . . 4 ⊢ (𝑛 ∈ ℕs → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
48		eqeq1 2739	. . . . 5 ⊢ (𝐴 = (𝐵𝐹( 1s /su 𝑛)) → (𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞)))))
49	48	2rexbidv 3206	. . . 4 ⊢ (𝐴 = (𝐵𝐹( 1s /su 𝑛)) → (∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞)))))
50	47, 49	syl5ibrcom 247	. . 3 ⊢ (𝑛 ∈ ℕs → (𝐴 = (𝐵𝐹( 1s /su 𝑛)) → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞)))))
51	50	rexlimiv 3134	. 2 ⊢ (∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)) → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
52	25, 51	impbii 209	1 ⊢ (∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060  (class class class)co 7405   No csur 27603   0_s c0s 27786   1_s c1s 27787   ·_s cmuls 28061   /_su cdivs 28142  ℕ_scnns 28259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-dc 10460

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-nadd 8678  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-0s 27788  df-1s 27789  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec 27897  df-norec2 27908  df-adds 27919  df-negs 27979  df-subs 27980  df-muls 28062  df-divs 28143  df-n0s 28260  df-nns 28261

This theorem is referenced by:  remulscl  28405