Theorem remulscllem1 28202

Description: Lemma for remulscl 28204. 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 7422	. . . . . . 7 ⊢ (𝑛 = (𝑝 ·s 𝑞) → ( 1s /su 𝑛) = ( 1s /su (𝑝 ·s 𝑞)))
2	1	oveq2d 7430	. . . . . 6 ⊢ (𝑛 = (𝑝 ·s 𝑞) → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
3	2	eqeq2d 2738	. . . . 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 28187	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝑝 ·s 𝑞) ∈ ℕs)
5		1sno 27734	. . . . . . . . 9 ⊢ 1s ∈ No
6	5	a1i 11	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 1s ∈ No )
7		nnsno 28170	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ∈ No )
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ∈ No )
9		nnsno 28170	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ∈ No )
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ∈ No )
11		nnne0s 28179	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ≠ 0s )
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ≠ 0s )
13		nnne0s 28179	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ≠ 0s )
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ≠ 0s )
15	6, 8, 6, 10, 12, 14	divmuldivsd 28104	. . . . . . 7 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)))
16		mulsrid 27987	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
17	5, 16	ax-mp 5	. . . . . . . 8 ⊢ ( 1s ·s 1s ) = 1s
18	17	oveq1i 7424	. . . . . . 7 ⊢ (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)) = ( 1s /su (𝑝 ·s 𝑞))
19	15, 18	eqtrdi 2783	. . . . . 6 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = ( 1s /su (𝑝 ·s 𝑞)))
20	19	oveq2d 7430	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
21	3, 4, 20	rspcedvdw 3610	. . . 4 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → ∃𝑛 ∈ ℕs (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛)))
22		eqeq1 2731	. . . . 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 3173	. . . 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 246	. . 3 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛))))
25	24	rexlimivv 3194	. 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 28170	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
28		nnne0s 28179	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
29	26, 27, 28	divscld 28096	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
30	29	mulsridd 27988	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (( 1s /su 𝑛) ·s 1s ) = ( 1s /su 𝑛))
31	30	eqcomd 2733	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) = (( 1s /su 𝑛) ·s 1s ))
32	31	oveq2d 7430	. . . . 5 ⊢ (𝑛 ∈ ℕs → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
33		1nns 28189	. . . . . 6 ⊢ 1s ∈ ℕs
34		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
35	34	oveq1d 7429	. . . . . . . . 9 ⊢ (𝑝 = 𝑛 → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑞)))
36	35	oveq2d 7430	. . . . . . . 8 ⊢ (𝑝 = 𝑛 → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))))
37	36	eqeq2d 2738	. . . . . . 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 7422	. . . . . . . . . . 11 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = ( 1s /su 1s ))
39		divs1 28077	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
40	5, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s /su 1s ) = 1s
41	38, 40	eqtrdi 2783	. . . . . . . . . 10 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = 1s )
42	41	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑞 = 1s → (( 1s /su 𝑛) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s 1s ))
43	42	oveq2d 7430	. . . . . . . 8 ⊢ (𝑞 = 1s → (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
44	43	eqeq2d 2738	. . . . . . 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 3620	. . . . . 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 1446	. . . . 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 686	. . . 4 ⊢ (𝑛 ∈ ℕs → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
48		eqeq1 2731	. . . . 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 3214	. . . 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 246	. . 3 ⊢ (𝑛 ∈ ℕs → (𝐴 = (𝐵𝐹( 1s /su 𝑛)) → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞)))))
51	50	rexlimiv 3143	. 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 208	1 ⊢ (∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∃wrex 3065  (class class class)co 7414   No csur 27547   0_s c0s 27729   1_s c1s 27730   ·_s cmuls 27980   /_su cdivs 28061  ℕ_scnns 28160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-dc 10455

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-nadd 8678  df-no 27550  df-slt 27551  df-bday 27552  df-sle 27652  df-sslt 27688  df-scut 27690  df-0s 27731  df-1s 27732  df-made 27748  df-old 27749  df-left 27751  df-right 27752  df-norec 27829  df-norec2 27840  df-adds 27851  df-negs 27908  df-subs 27909  df-muls 27981  df-divs 28062  df-n0s 28161  df-nns 28162

This theorem is referenced by:  remulscl  28204