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Theorem mulnqprl 6623
Description: Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
Assertion
Ref Expression
mulnqprl ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))

Proof of Theorem mulnqprl
Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 6456 . . . . . . 7 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
21adantl 262 . . . . . 6 (((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
3 simpr 103 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝑋Q)
4 prop 6530 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 6536 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → 𝐺Q)
64, 5sylan 267 . . . . . . . 8 ((𝐴P𝐺 ∈ (1st𝐴)) → 𝐺Q)
76ad2antrr 457 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐺Q)
8 prop 6530 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 elprnql 6536 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (1st𝐵)) → 𝐻Q)
108, 9sylan 267 . . . . . . . 8 ((𝐵P𝐻 ∈ (1st𝐵)) → 𝐻Q)
1110ad2antlr 458 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐻Q)
12 mulclnq 6431 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 ·Q 𝐻) ∈ Q)
137, 11, 12syl2anc 391 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 𝐻) ∈ Q)
14 recclnq 6447 . . . . . . 7 (𝐻Q → (*Q𝐻) ∈ Q)
1511, 14syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (*Q𝐻) ∈ Q)
16 mulcomnqg 6438 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
1716adantl 262 . . . . . 6 (((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
182, 3, 13, 15, 17caovord2d 5633 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻))))
19 mulassnqg 6439 . . . . . . . 8 ((𝐺Q𝐻Q ∧ (*Q𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
207, 11, 15, 19syl3anc 1135 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
21 recidnq 6448 . . . . . . . . 9 (𝐻Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5491 . . . . . . . 8 (𝐻Q → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
2311, 22syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
24 mulidnq 6444 . . . . . . . 8 (𝐺Q → (𝐺 ·Q 1Q) = 𝐺)
257, 24syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 1Q) = 𝐺)
2620, 23, 253eqtrd 2076 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = 𝐺)
2726breq2d 3773 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
2818, 27bitrd 177 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
29 prcdnql 6539 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
304, 29sylan 267 . . . . 5 ((𝐴P𝐺 ∈ (1st𝐴)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
3130ad2antrr 457 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
3228, 31sylbid 139 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
33 df-imp 6524 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}⟩)
34 mulclnq 6431 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
3533, 34genpprecll 6569 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) ∧ 𝐻 ∈ (1st𝐵)) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
3635exp4b 349 . . . . . . 7 (𝐴P → (𝐵P → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → (𝐻 ∈ (1st𝐵) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))))
3736com34 77 . . . . . 6 (𝐴P → (𝐵P → (𝐻 ∈ (1st𝐵) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))))
3837imp32 244 . . . . 5 ((𝐴P ∧ (𝐵P𝐻 ∈ (1st𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
3938adantlr 446 . . . 4 (((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
4039adantr 261 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
4132, 40syld 40 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
42 mulassnqg 6439 . . . . 5 ((𝑋Q ∧ (*Q𝐻) ∈ Q𝐻Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
433, 15, 11, 42syl3anc 1135 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
44 mulcomnqg 6438 . . . . . . 7 (((*Q𝐻) ∈ Q𝐻Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4515, 11, 44syl2anc 391 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4611, 21syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐻 ·Q (*Q𝐻)) = 1Q)
4745, 46eqtrd 2072 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = 1Q)
4847oveq2d 5491 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 6444 . . . . 5 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
5049adantl 262 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
5143, 48, 503eqtrd 2076 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = 𝑋)
5251eleq1d 2106 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
5341, 52sylibd 138 1 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  cop 3375   class class class wbr 3761  cfv 4865  (class class class)co 5475  1st c1st 5728  2nd c2nd 5729  Qcnq 6335  1Qc1q 6336   ·Q cmq 6338  *Qcrq 6339   <Q cltq 6340  Pcnp 6346   ·P cmp 6349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-1o 5964  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-mi 6361  df-lti 6362  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-inp 6521  df-imp 6524
This theorem is referenced by:  mullocprlem  6625  mulclpr  6627
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