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Theorem remulscllem1 28432
Description: Lemma for remulscl 28434. Split a product of reciprocals of naturals. (Contributed by Scott Fenton, 16-Apr-2025.)
Assertion
Ref Expression
remulscllem1 (∃𝑝 ∈ ℕs𝑞 ∈ 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
StepHypRef Expression
1 oveq2 7439 . . . . . . 7 (𝑛 = (𝑝 ·s 𝑞) → ( 1s /su 𝑛) = ( 1s /su (𝑝 ·s 𝑞)))
21oveq2d 7447 . . . . . 6 (𝑛 = (𝑝 ·s 𝑞) → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
32eqeq2d 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 28350 . . . . 5 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (𝑝 ·s 𝑞) ∈ ℕs)
5 1sno 27872 . . . . . . . . 9 1s No
65a1i 11 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 1s No )
7 nnsno 28329 . . . . . . . . 9 (𝑝 ∈ ℕs𝑝 No )
87adantr 480 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑝 No )
9 nnsno 28329 . . . . . . . . 9 (𝑞 ∈ ℕs𝑞 No )
109adantl 481 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑞 No )
11 nnne0s 28340 . . . . . . . . 9 (𝑝 ∈ ℕs𝑝 ≠ 0s )
1211adantr 480 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑝 ≠ 0s )
13 nnne0s 28340 . . . . . . . . 9 (𝑞 ∈ ℕs𝑞 ≠ 0s )
1413adantl 481 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑞 ≠ 0s )
156, 8, 6, 10, 12, 14divmuldivsd 28256 . . . . . . 7 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)))
16 mulsrid 28139 . . . . . . . . 9 ( 1s No → ( 1s ·s 1s ) = 1s )
175, 16ax-mp 5 . . . . . . . 8 ( 1s ·s 1s ) = 1s
1817oveq1i 7441 . . . . . . 7 (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)) = ( 1s /su (𝑝 ·s 𝑞))
1915, 18eqtrdi 2793 . . . . . 6 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = ( 1s /su (𝑝 ·s 𝑞)))
2019oveq2d 7447 . . . . 5 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
213, 4, 20rspcedvdw 3625 . . . 4 ((𝑝 ∈ ℕs𝑞 ∈ 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))))
2322rexbidv 3179 . . . 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))))
2421, 23syl5ibrcom 247 . . 3 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛))))
2524rexlimivv 3201 . 2 (∃𝑝 ∈ ℕs𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)))
265a1i 11 . . . . . . . . 9 (𝑛 ∈ ℕs → 1s No )
27 nnsno 28329 . . . . . . . . 9 (𝑛 ∈ ℕs𝑛 No )
28 nnne0s 28340 . . . . . . . . 9 (𝑛 ∈ ℕs𝑛 ≠ 0s )
2926, 27, 28divscld 28248 . . . . . . . 8 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
3029mulsridd 28140 . . . . . . 7 (𝑛 ∈ ℕs → (( 1s /su 𝑛) ·s 1s ) = ( 1s /su 𝑛))
3130eqcomd 2743 . . . . . 6 (𝑛 ∈ ℕs → ( 1s /su 𝑛) = (( 1s /su 𝑛) ·s 1s ))
3231oveq2d 7447 . . . . 5 (𝑛 ∈ ℕs → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
33 1nns 28352 . . . . . 6 1s ∈ ℕs
34 oveq2 7439 . . . . . . . . . 10 (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
3534oveq1d 7446 . . . . . . . . 9 (𝑝 = 𝑛 → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑞)))
3635oveq2d 7447 . . . . . . . 8 (𝑝 = 𝑛 → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))))
3736eqeq2d 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 7439 . . . . . . . . . . 11 (𝑞 = 1s → ( 1s /su 𝑞) = ( 1s /su 1s ))
39 divs1 28229 . . . . . . . . . . . 12 ( 1s No → ( 1s /su 1s ) = 1s )
405, 39ax-mp 5 . . . . . . . . . . 11 ( 1s /su 1s ) = 1s
4138, 40eqtrdi 2793 . . . . . . . . . 10 (𝑞 = 1s → ( 1s /su 𝑞) = 1s )
4241oveq2d 7447 . . . . . . . . 9 (𝑞 = 1s → (( 1s /su 𝑛) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s 1s ))
4342oveq2d 7447 . . . . . . . 8 (𝑞 = 1s → (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
4443eqeq2d 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 ))))
4537, 44rspc2ev 3635 . . . . . 6 ((𝑛 ∈ ℕs ∧ 1s ∈ ℕs ∧ (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s ))) → ∃𝑝 ∈ ℕs𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
4633, 45mp3an2 1451 . . . . 5 ((𝑛 ∈ ℕs ∧ (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s ))) → ∃𝑝 ∈ ℕs𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
4732, 46mpdan 687 . . . 4 (𝑛 ∈ ℕs → ∃𝑝 ∈ ℕs𝑞 ∈ 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)))))
49482rexbidv 3222 . . . 4 (𝐴 = (𝐵𝐹( 1s /su 𝑛)) → (∃𝑝 ∈ ℕs𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ ∃𝑝 ∈ ℕs𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞)))))
5047, 49syl5ibrcom 247 . . 3 (𝑛 ∈ ℕs → (𝐴 = (𝐵𝐹( 1s /su 𝑛)) → ∃𝑝 ∈ ℕs𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞)))))
5150rexlimiv 3148 . 2 (∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)) → ∃𝑝 ∈ ℕs𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
5225, 51impbii 209 1 (∃𝑝 ∈ ℕs𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) ↔ ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  (class class class)co 7431   No csur 27684   0s c0s 27867   1s c1s 27868   ·s cmuls 28132   /su cdivs 28213  scnns 28319
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-dc 10486
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214  df-n0s 28320  df-nns 28321
This theorem is referenced by:  remulscl  28434
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