Theorem remulscllem1 28447

Description: Lemma for remulscl 28449. 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 7439	. . . . . . 7 ⊢ (𝑛 = (𝑝 ·s 𝑞) → ( 1s /su 𝑛) = ( 1s /su (𝑝 ·s 𝑞)))
2	1	oveq2d 7447	. . . . . 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 28365	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝑝 ·s 𝑞) ∈ ℕs)
5		1sno 27887	. . . . . . . . 9 ⊢ 1s ∈ No
6	5	a1i 11	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 1s ∈ No )
7		nnsno 28344	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ∈ No )
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ∈ No )
9		nnsno 28344	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ∈ No )
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ∈ No )
11		nnne0s 28355	. . . . . . . . 9 ⊢ (𝑝 ∈ ℕs → 𝑝 ≠ 0s )
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑝 ≠ 0s )
13		nnne0s 28355	. . . . . . . . 9 ⊢ (𝑞 ∈ ℕs → 𝑞 ≠ 0s )
14	13	adantl 481	. . . . . . . 8 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → 𝑞 ≠ 0s )
15	6, 8, 6, 10, 12, 14	divmuldivsd 28271	. . . . . . 7 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)))
16		mulsrid 28154	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
17	5, 16	ax-mp 5	. . . . . . . 8 ⊢ ( 1s ·s 1s ) = 1s
18	17	oveq1i 7441	. . . . . . 7 ⊢ (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)) = ( 1s /su (𝑝 ·s 𝑞))
19	15, 18	eqtrdi 2791	. . . . . 6 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = ( 1s /su (𝑝 ·s 𝑞)))
20	19	oveq2d 7447	. . . . 5 ⊢ ((𝑝 ∈ ℕs ∧ 𝑞 ∈ ℕs) → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
21	3, 4, 20	rspcedvdw 3625	. . . 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 3177	. . . 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 3199	. 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 28344	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
28		nnne0s 28355	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
29	26, 27, 28	divscld 28263	. . . . . . . 8 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
30	29	mulsridd 28155	. . . . . . 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 7447	. . . . 5 ⊢ (𝑛 ∈ ℕs → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
33		1nns 28367	. . . . . 6 ⊢ 1s ∈ ℕs
34		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
35	34	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑝 = 𝑛 → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑞)))
36	35	oveq2d 7447	. . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = ( 1s /su 1s ))
39		divs1 28244	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
40	5, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s /su 1s ) = 1s
41	38, 40	eqtrdi 2791	. . . . . . . . . 10 ⊢ (𝑞 = 1s → ( 1s /su 𝑞) = 1s )
42	41	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑞 = 1s → (( 1s /su 𝑛) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s 1s ))
43	42	oveq2d 7447	. . . . . . . 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 3635	. . . . . 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 1448	. . . . 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 3220	. . . 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 3146	. 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 1537   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  (class class class)co 7431   No csur 27699   0_s c0s 27882   1_s c1s 27883   ·_s cmuls 28147   /_su cdivs 28228  ℕ_scnns 28334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229  df-n0s 28335  df-nns 28336

This theorem is referenced by:  remulscl  28449