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Theorem addnqprl 7349
Description: Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addnqprl ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))

Proof of Theorem addnqprl
Dummy variables 𝑟 𝑞 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7295 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 addnqprllem 7347 . . . . . 6 (((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴)))
31, 2sylanl1 399 . . . . 5 (((𝐴P𝐺 ∈ (1st𝐴)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴)))
43adantlr 468 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴)))
5 prop 7295 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 addnqprllem 7347 . . . . . 6 (((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (1st𝐵)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)))
75, 6sylanl1 399 . . . . 5 (((𝐵P𝐻 ∈ (1st𝐵)) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)))
87adantll 467 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)))
94, 8jcad 305 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵))))
10 simpl 108 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))))
11 simpl 108 . . . . 5 ((𝐴P𝐺 ∈ (1st𝐴)) → 𝐴P)
12 simpl 108 . . . . 5 ((𝐵P𝐻 ∈ (1st𝐵)) → 𝐵P)
1311, 12anim12i 336 . . . 4 (((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) → (𝐴P𝐵P))
14 df-iplp 7288 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (1st𝑥) ∧ 𝑠 ∈ (1st𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞Q ∣ ∃𝑟Q𝑠Q (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (2nd𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
15 addclnq 7195 . . . . 5 ((𝑟Q𝑠Q) → (𝑟 +Q 𝑠) ∈ Q)
1614, 15genpprecll 7334 . . . 4 ((𝐴P𝐵P) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵))))
1710, 13, 163syl 17 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (1st𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (1st𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵))))
189, 17syld 45 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 +Q 𝐻) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵))))
19 simpr 109 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝑋Q)
20 elprnql 7301 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → 𝐺Q)
211, 20sylan 281 . . . . . . . 8 ((𝐴P𝐺 ∈ (1st𝐴)) → 𝐺Q)
2221ad2antrr 479 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐺Q)
23 elprnql 7301 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (1st𝐵)) → 𝐻Q)
245, 23sylan 281 . . . . . . . 8 ((𝐵P𝐻 ∈ (1st𝐵)) → 𝐻Q)
2524ad2antlr 480 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐻Q)
26 addclnq 7195 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 +Q 𝐻) ∈ Q)
2722, 25, 26syl2anc 408 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 +Q 𝐻) ∈ Q)
28 recclnq 7212 . . . . . 6 ((𝐺 +Q 𝐻) ∈ Q → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
2927, 28syl 14 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
30 mulassnqg 7204 . . . . 5 ((𝑋Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
3119, 29, 27, 30syl3anc 1216 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
32 mulclnq 7196 . . . . . 6 ((𝑋Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
3319, 29, 32syl2anc 408 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
34 distrnqg 7207 . . . . 5 (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q𝐺Q𝐻Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
3533, 22, 25, 34syl3anc 1216 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
36 mulcomnqg 7203 . . . . . . . 8 (((*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
3729, 27, 36syl2anc 408 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
38 recidnq 7213 . . . . . . . 8 ((𝐺 +Q 𝐻) ∈ Q → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
3927, 38syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
4037, 39eqtrd 2172 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = 1Q)
4140oveq2d 5790 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = (𝑋 ·Q 1Q))
42 mulidnq 7209 . . . . . 6 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
4342adantl 275 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
4441, 43eqtrd 2172 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = 𝑋)
4531, 35, 443eqtr3d 2180 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) = 𝑋)
4645eleq1d 2208 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (1st ‘(𝐴 +P 𝐵)) ↔ 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))
4718, 46sylibd 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   = wceq 1331  wcel 1480  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100  1Qc1q 7101   +Q cplq 7102   ·Q cmq 7103  *Qcrq 7104   <Q cltq 7105  Pcnp 7111   +P cpp 7113
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-inp 7286  df-iplp 7288
This theorem is referenced by:  addlocprlemlt  7351  addclpr  7357
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