Theorem remulscllem1 28351

Description: Lemma for remulscl 28353. 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 7395	. . . . . . 7 ⊢ (𝑛 = (𝑝 ·s 𝑞) → ( 1s /su 𝑛) = ( 1s /su (𝑝 ·s 𝑞)))
2	1	oveq2d 7403	. . . . . 6 ⊢ (𝑛 = (𝑝 ·s 𝑞) → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
3	2	eqeq2d 2740	. . . . 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 28239	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝑝 ·s 𝑞) ∈ ℕs)
5		1sno 27739	. . . . . . . . 9 ⊢ 1s ∈ No
6	5	a1i 11	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 1s ∈ No )
7		nnsno 28217	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ∈ No )
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ∈ No )
9		nnsno 28217	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ∈ No )
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ∈ No )
11		nnne0s 28229	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ≠ 0s )
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ≠ 0s )
13		nnne0s 28229	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ≠ 0s )
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ≠ 0s )
15	6, 8, 6, 10, 12, 14	divmuldivsd 28134	. . . . . . 7 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)))
16		mulsrid 28016	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
17	5, 16	ax-mp 5	. . . . . . . 8 ⊢ ( 1s ·s 1s ) = 1s
18	17	oveq1i 7397	. . . . . . 7 ⊢ (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)) = ( 1s /su (𝑝 ·s 𝑞))
19	15, 18	eqtrdi 2780	. . . . . 6 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = ( 1s /su (𝑝 ·s 𝑞)))
20	19	oveq2d 7403	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
21	3, 4, 20	rspcedvdw 3591	. . . 4 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → ∃𝑛 ∈ ℕs (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛)))
22		eqeq1 2733	. . . . 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 3157	. . . 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 3179	. 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 28217	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
28		nnne0s 28229	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
29	26, 27, 28	divscld 28126	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
30	29	mulsridd 28017	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (( 1s /su 𝑛) ·s 1s ) = ( 1s /su 𝑛))
31	30	eqcomd 2735	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) = (( 1s /su 𝑛) ·s 1s ))
32	31	oveq2d 7403	. . . . 5 ⊢ (𝑛 ∈ ℕs → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
33		1nns 28241	. . . . . 6 ⊢ 1s ∈ ℕs
34		oveq2 7395	. . . . . . . . . 10 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
35	34	oveq1d 7402	. . . . . . . . 9 ⊢ (𝑝 = 𝑛 → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑞)))
36	35	oveq2d 7403	. . . . . . . 8 ⊢ (𝑝 = 𝑛 → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))))
37	36	eqeq2d 2740	. . . . . . 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 7395	. . . . . . . . . . 11 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = ( 1s /su 1s ))
39		divs1 28107	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
40	5, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s /su 1s ) = 1s
41	38, 40	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = 1s )
42	41	oveq2d 7403	. . . . . . . . 9 ⊢ (𝑞 = 1s → (( 1s /su 𝑛) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s 1s ))
43	42	oveq2d 7403	. . . . . . . 8 ⊢ (𝑞 = 1s → (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
44	43	eqeq2d 2740	. . . . . . 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 3601	. . . . . 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 2733	. . . . 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 3202	. . . 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 3127	. 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 2109   ≠ wne 2925  ∃wrex 3053  (class class class)co 7387   No csur 27551   0_s c0s 27734   1_s c1s 27735   ·_s cmuls 28009   /_su cdivs 28090  ℕ_scnns 28207

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-dc 10399

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-norec2 27856  df-adds 27867  df-negs 27927  df-subs 27928  df-muls 28010  df-divs 28091  df-n0s 28208  df-nns 28209

This theorem is referenced by:  remulscl  28353