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Theorem mulnqprl 6556
Description: Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
Assertion
Ref Expression
mulnqprl ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (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 6389 . . . . . . 7 ((y Q z Q w Q) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
21adantl 262 . . . . . 6 (((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) (y Q z Q w Q)) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
3 simpr 103 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝑋 Q)
4 prop 6463 . . . . . . . . 9 (A P → ⟨(1stA), (2ndA)⟩ P)
5 elprnql 6469 . . . . . . . . 9 ((⟨(1stA), (2ndA)⟩ P 𝐺 (1stA)) → 𝐺 Q)
64, 5sylan 267 . . . . . . . 8 ((A P 𝐺 (1stA)) → 𝐺 Q)
76ad2antrr 457 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝐺 Q)
8 prop 6463 . . . . . . . . 9 (B P → ⟨(1stB), (2ndB)⟩ P)
9 elprnql 6469 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P 𝐻 (1stB)) → 𝐻 Q)
108, 9sylan 267 . . . . . . . 8 ((B P 𝐻 (1stB)) → 𝐻 Q)
1110ad2antlr 458 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝐻 Q)
12 mulclnq 6364 . . . . . . 7 ((𝐺 Q 𝐻 Q) → (𝐺 ·Q 𝐻) Q)
137, 11, 12syl2anc 391 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐺 ·Q 𝐻) Q)
14 recclnq 6380 . . . . . . 7 (𝐻 Q → (*Q𝐻) Q)
1511, 14syl 14 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (*Q𝐻) Q)
16 mulcomnqg 6371 . . . . . . 7 ((y Q z Q) → (y ·Q z) = (z ·Q y))
1716adantl 262 . . . . . 6 (((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) (y Q z Q)) → (y ·Q z) = (z ·Q y))
182, 3, 13, 15, 17caovord2d 5615 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻))))
19 mulassnqg 6372 . . . . . . . 8 ((𝐺 Q 𝐻 Q (*Q𝐻) Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
207, 11, 15, 19syl3anc 1135 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
21 recidnq 6381 . . . . . . . . 9 (𝐻 Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5474 . . . . . . . 8 (𝐻 Q → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
2311, 22syl 14 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
24 mulidnq 6377 . . . . . . . 8 (𝐺 Q → (𝐺 ·Q 1Q) = 𝐺)
257, 24syl 14 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐺 ·Q 1Q) = 𝐺)
2620, 23, 253eqtrd 2076 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = 𝐺)
2726breq2d 3770 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
2818, 27bitrd 177 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
29 prcdnql 6472 . . . . . 6 ((⟨(1stA), (2ndA)⟩ P 𝐺 (1stA)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) (1stA)))
304, 29sylan 267 . . . . 5 ((A P 𝐺 (1stA)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) (1stA)))
3130ad2antrr 457 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) (1stA)))
3228, 31sylbid 139 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → (𝑋 ·Q (*Q𝐻)) (1stA)))
33 df-imp 6457 . . . . . . . . 9 ·P = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y ·Q z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y ·Q z))}⟩)
34 mulclnq 6364 . . . . . . . . 9 ((y Q z Q) → (y ·Q z) Q)
3533, 34genpprecll 6502 . . . . . . . 8 ((A P B P) → (((𝑋 ·Q (*Q𝐻)) (1stA) 𝐻 (1stB)) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
3635exp4b 349 . . . . . . 7 (A P → (B P → ((𝑋 ·Q (*Q𝐻)) (1stA) → (𝐻 (1stB) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))))
3736com34 77 . . . . . 6 (A P → (B P → (𝐻 (1stB) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))))
3837imp32 244 . . . . 5 ((A P (B P 𝐻 (1stB))) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
3938adantlr 446 . . . 4 (((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
4039adantr 261 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
4132, 40syld 40 . 2 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
42 mulassnqg 6372 . . . . 5 ((𝑋 Q (*Q𝐻) Q 𝐻 Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
433, 15, 11, 42syl3anc 1135 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
44 mulcomnqg 6371 . . . . . . 7 (((*Q𝐻) Q 𝐻 Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4515, 11, 44syl2anc 391 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4611, 21syl 14 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐻 ·Q (*Q𝐻)) = 1Q)
4745, 46eqtrd 2072 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((*Q𝐻) ·Q 𝐻) = 1Q)
4847oveq2d 5474 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 6377 . . . . 5 (𝑋 Q → (𝑋 ·Q 1Q) = 𝑋)
5049adantl 262 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q 1Q) = 𝑋)
5143, 48, 503eqtrd 2076 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = 𝑋)
5251eleq1d 2106 . 2 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B)) ↔ 𝑋 (1st ‘(A ·P B))))
5341, 52sylibd 138 1 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (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 3373   class class class wbr 3758  cfv 4848  (class class class)co 5458  1st c1st 5710  2nd c2nd 5711  Qcnq 6268  1Qc1q 6269   ·Q cmq 6271  *Qcrq 6272   <Q cltq 6273  Pcnp 6279   ·P cmp 6282
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 3866  ax-sep 3869  ax-nul 3877  ax-pow 3921  ax-pr 3938  ax-un 4139  ax-setind 4223  ax-iinf 4257
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 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-int 3610  df-iun 3653  df-br 3759  df-opab 3813  df-mpt 3814  df-tr 3849  df-eprel 4020  df-id 4024  df-iord 4072  df-on 4074  df-suc 4077  df-iom 4260  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-rn 4302  df-res 4303  df-ima 4304  df-iota 4813  df-fun 4850  df-fn 4851  df-f 4852  df-f1 4853  df-fo 4854  df-f1o 4855  df-fv 4856  df-ov 5461  df-oprab 5462  df-mpt2 5463  df-1st 5712  df-2nd 5713  df-recs 5865  df-irdg 5901  df-1o 5944  df-oadd 5948  df-omul 5949  df-er 6046  df-ec 6048  df-qs 6052  df-ni 6292  df-mi 6294  df-lti 6295  df-mpq 6333  df-enq 6335  df-nqqs 6336  df-mqqs 6338  df-1nqqs 6339  df-rq 6340  df-ltnqqs 6341  df-inp 6454  df-imp 6457
This theorem is referenced by:  mullocprlem  6558  mulclpr  6560
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