Theorem remulscllem1 28508

Description: Lemma for remulscl 28510. 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 7376	. . . . . . 7 ⊢ (𝑛 = (𝑝 ·s 𝑞) → ( 1s /su 𝑛) = ( 1s /su (𝑝 ·s 𝑞)))
2	1	oveq2d 7384	. . . . . 6 ⊢ (𝑛 = (𝑝 ·s 𝑞) → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
3	2	eqeq2d 2748	. . . . 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 28355	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝑝 ·s 𝑞) ∈ ℕs)
5		1no 27818	. . . . . . . . 9 ⊢ 1s ∈ No
6	5	a1i 11	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 1s ∈ No )
7		nnno 28332	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ∈ No )
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ∈ No )
9		nnno 28332	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ∈ No )
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ∈ No )
11		nnne0s 28345	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ≠ 0s )
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ≠ 0s )
13		nnne0s 28345	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ≠ 0s )
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ≠ 0s )
15	6, 8, 6, 10, 12, 14	divmuldivsd 28240	. . . . . . 7 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)))
16		mulsrid 28121	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
17	5, 16	ax-mp 5	. . . . . . . 8 ⊢ ( 1s ·s 1s ) = 1s
18	17	oveq1i 7378	. . . . . . 7 ⊢ (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)) = ( 1s /su (𝑝 ·s 𝑞))
19	15, 18	eqtrdi 2788	. . . . . 6 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = ( 1s /su (𝑝 ·s 𝑞)))
20	19	oveq2d 7384	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
21	3, 4, 20	rspcedvdw 3581	. . . 4 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → ∃𝑛 ∈ ℕs (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛)))
22		eqeq1 2741	. . . . 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 3162	. . . 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 3180	. 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		nnno 28332	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
28		nnne0s 28345	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
29	26, 27, 28	divscld 28232	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
30	29	mulsridd 28122	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → (( 1s /su 𝑛) ·s 1s ) = ( 1s /su 𝑛))
31	30	eqcomd 2743	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) = (( 1s /su 𝑛) ·s 1s ))
32	31	oveq2d 7384	. . . . 5 ⊢ (𝑛 ∈ ℕs → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
33		1nns 28357	. . . . . 6 ⊢ 1s ∈ ℕs
34		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
35	34	oveq1d 7383	. . . . . . . . 9 ⊢ (𝑝 = 𝑛 → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑞)))
36	35	oveq2d 7384	. . . . . . . 8 ⊢ (𝑝 = 𝑛 → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))))
37	36	eqeq2d 2748	. . . . . . 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 7376	. . . . . . . . . . 11 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = ( 1s /su 1s ))
39		divs1 28212	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
40	5, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s /su 1s ) = 1s
41	38, 40	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = 1s )
42	41	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑞 = 1s → (( 1s /su 𝑛) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s 1s ))
43	42	oveq2d 7384	. . . . . . . 8 ⊢ (𝑞 = 1s → (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
44	43	eqeq2d 2748	. . . . . . 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 3591	. . . . . 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 1452	. . . . 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 688	. . . 4 ⊢ (𝑛 ∈ ℕs → ∃𝑝 ∈ ℕs ∃𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
48		eqeq1 2741	. . . . 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 3203	. . . 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 3132	. 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 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  (class class class)co 7368   No csur 27619   0_s c0s 27813   1_s c1s 27814   ·_s cmuls 28114   /_su cdivs 28195  ℕ_scnns 28321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-dc 10368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115  df-divs 28196  df-n0s 28322  df-nns 28323

This theorem is referenced by:  remulscl  28510