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Theorem mulnqprl 7523
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 7356 . . . . . . 7 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
21adantl 275 . . . . . 6 (((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
3 simpr 109 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝑋Q)
4 prop 7430 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 7436 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → 𝐺Q)
64, 5sylan 281 . . . . . . . 8 ((𝐴P𝐺 ∈ (1st𝐴)) → 𝐺Q)
76ad2antrr 485 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐺Q)
8 prop 7430 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 elprnql 7436 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (1st𝐵)) → 𝐻Q)
108, 9sylan 281 . . . . . . . 8 ((𝐵P𝐻 ∈ (1st𝐵)) → 𝐻Q)
1110ad2antlr 486 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐻Q)
12 mulclnq 7331 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 ·Q 𝐻) ∈ Q)
137, 11, 12syl2anc 409 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 𝐻) ∈ Q)
14 recclnq 7347 . . . . . . 7 (𝐻Q → (*Q𝐻) ∈ Q)
1511, 14syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (*Q𝐻) ∈ Q)
16 mulcomnqg 7338 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
1716adantl 275 . . . . . 6 (((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
182, 3, 13, 15, 17caovord2d 6020 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻))))
19 mulassnqg 7339 . . . . . . . 8 ((𝐺Q𝐻Q ∧ (*Q𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
207, 11, 15, 19syl3anc 1233 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
21 recidnq 7348 . . . . . . . . 9 (𝐻Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5867 . . . . . . . 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 7344 . . . . . . . 8 (𝐺Q → (𝐺 ·Q 1Q) = 𝐺)
257, 24syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 1Q) = 𝐺)
2620, 23, 253eqtrd 2207 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = 𝐺)
2726breq2d 3999 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
2818, 27bitrd 187 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
29 prcdnql 7439 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
304, 29sylan 281 . . . . 5 ((𝐴P𝐺 ∈ (1st𝐴)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
3130ad2antrr 485 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
3228, 31sylbid 149 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
33 df-imp 7424 . . . . . . . . 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 7331 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
3533, 34genpprecll 7469 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) ∧ 𝐻 ∈ (1st𝐵)) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
3635exp4b 365 . . . . . . 7 (𝐴P → (𝐵P → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → (𝐻 ∈ (1st𝐵) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))))
3736com34 83 . . . . . 6 (𝐴P → (𝐵P → (𝐻 ∈ (1st𝐵) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))))
3837imp32 255 . . . . 5 ((𝐴P ∧ (𝐵P𝐻 ∈ (1st𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
3938adantlr 474 . . . 4 (((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
4039adantr 274 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
4132, 40syld 45 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
42 mulassnqg 7339 . . . . 5 ((𝑋Q ∧ (*Q𝐻) ∈ Q𝐻Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
433, 15, 11, 42syl3anc 1233 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
44 mulcomnqg 7338 . . . . . . 7 (((*Q𝐻) ∈ Q𝐻Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4515, 11, 44syl2anc 409 . . . . . 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 2203 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = 1Q)
4847oveq2d 5867 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 7344 . . . . 5 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
5049adantl 275 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
5143, 48, 503eqtrd 2207 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = 𝑋)
5251eleq1d 2239 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
5341, 52sylibd 148 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 103  wb 104  w3a 973   = wceq 1348  wcel 2141  cop 3584   class class class wbr 3987  cfv 5196  (class class class)co 5851  1st c1st 6115  2nd c2nd 6116  Qcnq 7235  1Qc1q 7236   ·Q cmq 7238  *Qcrq 7239   <Q cltq 7240  Pcnp 7246   ·P cmp 7249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-oadd 6397  df-omul 6398  df-er 6511  df-ec 6513  df-qs 6517  df-ni 7259  df-mi 7261  df-lti 7262  df-mpq 7300  df-enq 7302  df-nqqs 7303  df-mqqs 7305  df-1nqqs 7306  df-rq 7307  df-ltnqqs 7308  df-inp 7421  df-imp 7424
This theorem is referenced by:  mullocprlem  7525  mulclpr  7527
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