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Theorem mulclpr 6543
Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
Assertion
Ref Expression
mulclpr ((A P B P) → (A ·P B) P)

Proof of Theorem mulclpr
Dummy variables 𝑞 𝑟 𝑡 u v w x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-imp 6444 . . . 4 ·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))}⟩)
21genpelxp 6486 . . 3 ((A P B P) → (A ·P B) (𝒫 Q × 𝒫 Q))
3 mulclnq 6353 . . . 4 ((y Q z Q) → (y ·Q z) Q)
41, 3genpml 6493 . . 3 ((A P B P) → 𝑞 Q 𝑞 (1st ‘(A ·P B)))
51, 3genpmu 6494 . . 3 ((A P B P) → 𝑟 Q 𝑟 (2nd ‘(A ·P B)))
62, 4, 5jca32 293 . 2 ((A P B P) → ((A ·P B) (𝒫 Q × 𝒫 Q) (𝑞 Q 𝑞 (1st ‘(A ·P B)) 𝑟 Q 𝑟 (2nd ‘(A ·P B)))))
7 ltmnqg 6378 . . . . 5 ((x Q y Q z Q) → (x <Q y ↔ (z ·Q x) <Q (z ·Q y)))
8 mulcomnqg 6360 . . . . 5 ((x Q y Q) → (x ·Q y) = (y ·Q x))
9 mulnqprl 6539 . . . . 5 ((((A P u (1stA)) (B P 𝑡 (1stB))) x Q) → (x <Q (u ·Q 𝑡) → x (1st ‘(A ·P B))))
101, 3, 7, 8, 9genprndl 6497 . . . 4 ((A P B P) → 𝑞 Q (𝑞 (1st ‘(A ·P B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A ·P B)))))
11 mulnqpru 6540 . . . . 5 ((((A P u (2ndA)) (B P 𝑡 (2ndB))) x Q) → ((u ·Q 𝑡) <Q xx (2nd ‘(A ·P B))))
121, 3, 7, 8, 11genprndu 6498 . . . 4 ((A P B P) → 𝑟 Q (𝑟 (2nd ‘(A ·P B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A ·P B)))))
1310, 12jca 290 . . 3 ((A P B P) → (𝑞 Q (𝑞 (1st ‘(A ·P B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A ·P B)))) 𝑟 Q (𝑟 (2nd ‘(A ·P B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A ·P B))))))
141, 3, 7, 8genpdisj 6499 . . 3 ((A P B P) → 𝑞 Q ¬ (𝑞 (1st ‘(A ·P B)) 𝑞 (2nd ‘(A ·P B))))
15 mullocpr 6542 . . 3 ((A P B P) → 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A ·P B)) 𝑟 (2nd ‘(A ·P B)))))
1613, 14, 153jca 1083 . 2 ((A P B P) → ((𝑞 Q (𝑞 (1st ‘(A ·P B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A ·P B)))) 𝑟 Q (𝑟 (2nd ‘(A ·P B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A ·P B))))) 𝑞 Q ¬ (𝑞 (1st ‘(A ·P B)) 𝑞 (2nd ‘(A ·P B))) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A ·P B)) 𝑟 (2nd ‘(A ·P B))))))
17 elnp1st2nd 6451 . 2 ((A ·P B) P ↔ (((A ·P B) (𝒫 Q × 𝒫 Q) (𝑞 Q 𝑞 (1st ‘(A ·P B)) 𝑟 Q 𝑟 (2nd ‘(A ·P B)))) ((𝑞 Q (𝑞 (1st ‘(A ·P B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A ·P B)))) 𝑟 Q (𝑟 (2nd ‘(A ·P B)) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd ‘(A ·P B))))) 𝑞 Q ¬ (𝑞 (1st ‘(A ·P B)) 𝑞 (2nd ‘(A ·P B))) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A ·P B)) 𝑟 (2nd ‘(A ·P B)))))))
186, 16, 17sylanbrc 394 1 ((A P B P) → (A ·P B) P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   wcel 1390  wral 2300  wrex 2301  𝒫 cpw 3350   class class class wbr 3754   × cxp 4285  cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   ·Q cmq 6260   <Q cltq 6262  Pcnp 6268   ·P cmp 6271
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-2o 5934  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-enq0 6399  df-nq0 6400  df-0nq0 6401  df-plq0 6402  df-mq0 6403  df-inp 6441  df-imp 6444
This theorem is referenced by:  mulassprg  6547  distrlem1prl  6548  distrlem1pru  6549  distrlem4prl  6550  distrlem4pru  6551  distrlem5prl  6552  distrlem5pru  6553  distrprg  6554  1idpr  6558  recexprlemex  6599  ltmprr  6604  mulcmpblnrlemg  6620  mulcmpblnr  6621  mulclsr  6634  mulcomsrg  6637  mulasssrg  6638  distrsrg  6639  m1m1sr  6641  1idsr  6648  00sr  6649  recexgt0sr  6653  mulgt0sr  6656  mulextsr1lem  6658  mulextsr1  6659
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