MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  remulscllem1 Structured version   Visualization version   GIF version

Theorem remulscllem1 28227
Description: Lemma for remulscl 28229. 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 7428 . . . . . . 7 (𝑛 = (𝑝 ·s 𝑞) → ( 1s /su 𝑛) = ( 1s /su (𝑝 ·s 𝑞)))
21oveq2d 7436 . . . . . 6 (𝑛 = (𝑝 ·s 𝑞) → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
32eqeq2d 2739 . . . . 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 28212 . . . . 5 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (𝑝 ·s 𝑞) ∈ ℕs)
5 1sno 27759 . . . . . . . . 9 1s No
65a1i 11 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 1s No )
7 nnsno 28195 . . . . . . . . 9 (𝑝 ∈ ℕs𝑝 No )
87adantr 480 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑝 No )
9 nnsno 28195 . . . . . . . . 9 (𝑞 ∈ ℕs𝑞 No )
109adantl 481 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑞 No )
11 nnne0s 28204 . . . . . . . . 9 (𝑝 ∈ ℕs𝑝 ≠ 0s )
1211adantr 480 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑝 ≠ 0s )
13 nnne0s 28204 . . . . . . . . 9 (𝑞 ∈ ℕs𝑞 ≠ 0s )
1413adantl 481 . . . . . . . 8 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → 𝑞 ≠ 0s )
156, 8, 6, 10, 12, 14divmuldivsd 28129 . . . . . . 7 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)))
16 mulsrid 28012 . . . . . . . . 9 ( 1s No → ( 1s ·s 1s ) = 1s )
175, 16ax-mp 5 . . . . . . . 8 ( 1s ·s 1s ) = 1s
1817oveq1i 7430 . . . . . . 7 (( 1s ·s 1s ) /su (𝑝 ·s 𝑞)) = ( 1s /su (𝑝 ·s 𝑞))
1915, 18eqtrdi 2784 . . . . . 6 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = ( 1s /su (𝑝 ·s 𝑞)))
2019oveq2d 7436 . . . . 5 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su (𝑝 ·s 𝑞))))
213, 4, 20rspcedvdw 3612 . . . 4 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → ∃𝑛 ∈ ℕs (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹( 1s /su 𝑛)))
22 eqeq1 2732 . . . . 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 3175 . . . 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 246 . . 3 ((𝑝 ∈ ℕs𝑞 ∈ ℕs) → (𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) → ∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛))))
2524rexlimivv 3196 . 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 28195 . . . . . . . . 9 (𝑛 ∈ ℕs𝑛 No )
28 nnne0s 28204 . . . . . . . . 9 (𝑛 ∈ ℕs𝑛 ≠ 0s )
2926, 27, 28divscld 28121 . . . . . . . 8 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
3029mulsridd 28013 . . . . . . 7 (𝑛 ∈ ℕs → (( 1s /su 𝑛) ·s 1s ) = ( 1s /su 𝑛))
3130eqcomd 2734 . . . . . 6 (𝑛 ∈ ℕs → ( 1s /su 𝑛) = (( 1s /su 𝑛) ·s 1s ))
3231oveq2d 7436 . . . . 5 (𝑛 ∈ ℕs → (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
33 1nns 28214 . . . . . 6 1s ∈ ℕs
34 oveq2 7428 . . . . . . . . . 10 (𝑝 = 𝑛 → ( 1s /su 𝑝) = ( 1s /su 𝑛))
3534oveq1d 7435 . . . . . . . . 9 (𝑝 = 𝑛 → (( 1s /su 𝑝) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s ( 1s /su 𝑞)))
3635oveq2d 7436 . . . . . . . 8 (𝑝 = 𝑛 → (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))))
3736eqeq2d 2739 . . . . . . 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 7428 . . . . . . . . . . 11 (𝑞 = 1s → ( 1s /su 𝑞) = ( 1s /su 1s ))
39 divs1 28102 . . . . . . . . . . . 12 ( 1s No → ( 1s /su 1s ) = 1s )
405, 39ax-mp 5 . . . . . . . . . . 11 ( 1s /su 1s ) = 1s
4138, 40eqtrdi 2784 . . . . . . . . . 10 (𝑞 = 1s → ( 1s /su 𝑞) = 1s )
4241oveq2d 7436 . . . . . . . . 9 (𝑞 = 1s → (( 1s /su 𝑛) ·s ( 1s /su 𝑞)) = (( 1s /su 𝑛) ·s 1s ))
4342oveq2d 7436 . . . . . . . 8 (𝑞 = 1s → (𝐵𝐹(( 1s /su 𝑛) ·s ( 1s /su 𝑞))) = (𝐵𝐹(( 1s /su 𝑛) ·s 1s )))
4443eqeq2d 2739 . . . . . . 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 3622 . . . . . 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 1446 . . . . 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 686 . . . 4 (𝑛 ∈ ℕs → ∃𝑝 ∈ ℕs𝑞 ∈ ℕs (𝐵𝐹( 1s /su 𝑛)) = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
48 eqeq1 2732 . . . . 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 3216 . . . 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 246 . . 3 (𝑛 ∈ ℕs → (𝐴 = (𝐵𝐹( 1s /su 𝑛)) → ∃𝑝 ∈ ℕs𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞)))))
5150rexlimiv 3145 . 2 (∃𝑛 ∈ ℕs 𝐴 = (𝐵𝐹( 1s /su 𝑛)) → ∃𝑝 ∈ ℕs𝑞 ∈ ℕs 𝐴 = (𝐵𝐹(( 1s /su 𝑝) ·s ( 1s /su 𝑞))))
5225, 51impbii 208 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 205  wa 395   = wceq 1534  wcel 2099  wne 2937  wrex 3067  (class class class)co 7420   No csur 27572   0s c0s 27754   1s c1s 27755   ·s cmuls 28005   /su cdivs 28086  scnns 28185
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 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-dc 10469
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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-oadd 8490  df-nadd 8686  df-no 27575  df-slt 27576  df-bday 27577  df-sle 27677  df-sslt 27713  df-scut 27715  df-0s 27756  df-1s 27757  df-made 27773  df-old 27774  df-left 27776  df-right 27777  df-norec 27854  df-norec2 27865  df-adds 27876  df-negs 27933  df-subs 27934  df-muls 28006  df-divs 28087  df-n0s 28186  df-nns 28187
This theorem is referenced by:  remulscl  28229
  Copyright terms: Public domain W3C validator