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Theorem distrlem4prl 6560
 Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem4prl (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
Distinct variable groups:   x,y,z,f,A   x,B,y,z,f   x,𝐶,y,z,f

Proof of Theorem distrlem4prl
Dummy variables w v u g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 6385 . . . . . . 7 ((w Q v Q u Q) → (w <Q v ↔ (u ·Q w) <Q (u ·Q v)))
21adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q u Q)) → (w <Q v ↔ (u ·Q w) <Q (u ·Q v)))
3 simp1 903 . . . . . . 7 ((A P B P 𝐶 P) → A P)
4 simpll 481 . . . . . . 7 (((x (1stA) y (1stB)) (f (1stA) z (1st𝐶))) → x (1stA))
5 prop 6458 . . . . . . . 8 (A P → ⟨(1stA), (2ndA)⟩ P)
6 elprnql 6464 . . . . . . . 8 ((⟨(1stA), (2ndA)⟩ P x (1stA)) → x Q)
75, 6sylan 267 . . . . . . 7 ((A P x (1stA)) → x Q)
83, 4, 7syl2an 273 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → x Q)
9 simprl 483 . . . . . . 7 (((x (1stA) y (1stB)) (f (1stA) z (1st𝐶))) → f (1stA))
10 elprnql 6464 . . . . . . . 8 ((⟨(1stA), (2ndA)⟩ P f (1stA)) → f Q)
115, 10sylan 267 . . . . . . 7 ((A P f (1stA)) → f Q)
123, 9, 11syl2an 273 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → f Q)
13 simpl2 907 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → B P)
14 simprlr 490 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → y (1stB))
15 prop 6458 . . . . . . . 8 (B P → ⟨(1stB), (2ndB)⟩ P)
16 elprnql 6464 . . . . . . . 8 ((⟨(1stB), (2ndB)⟩ P y (1stB)) → y Q)
1715, 16sylan 267 . . . . . . 7 ((B P y (1stB)) → y Q)
1813, 14, 17syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → y Q)
19 mulcomnqg 6367 . . . . . . 7 ((w Q v Q) → (w ·Q v) = (v ·Q w))
2019adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q)) → (w ·Q v) = (v ·Q w))
212, 8, 12, 18, 20caovord2d 5612 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x <Q f ↔ (x ·Q y) <Q (f ·Q y)))
22 ltanqg 6384 . . . . . . 7 ((w Q v Q u Q) → (w <Q v ↔ (u +Q w) <Q (u +Q v)))
2322adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q u Q)) → (w <Q v ↔ (u +Q w) <Q (u +Q v)))
24 mulclnq 6360 . . . . . . 7 ((x Q y Q) → (x ·Q y) Q)
258, 18, 24syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q y) Q)
26 mulclnq 6360 . . . . . . 7 ((f Q y Q) → (f ·Q y) Q)
2712, 18, 26syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q y) Q)
28 simpl3 908 . . . . . . . 8 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → 𝐶 P)
29 simprrr 492 . . . . . . . 8 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → z (1st𝐶))
30 prop 6458 . . . . . . . . 9 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
31 elprnql 6464 . . . . . . . . 9 ((⟨(1st𝐶), (2nd𝐶)⟩ P z (1st𝐶)) → z Q)
3230, 31sylan 267 . . . . . . . 8 ((𝐶 P z (1st𝐶)) → z Q)
3328, 29, 32syl2anc 391 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → z Q)
34 mulclnq 6360 . . . . . . 7 ((f Q z Q) → (f ·Q z) Q)
3512, 33, 34syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q z) Q)
36 addcomnqg 6365 . . . . . . 7 ((w Q v Q) → (w +Q v) = (v +Q w))
3736adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q)) → (w +Q v) = (v +Q w))
3823, 25, 27, 35, 37caovord2d 5612 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) <Q (f ·Q y) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
3921, 38bitrd 177 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x <Q f ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
40 simpl1 906 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → A P)
41 addclpr 6520 . . . . . . . 8 ((B P 𝐶 P) → (B +P 𝐶) P)
42413adant1 921 . . . . . . 7 ((A P B P 𝐶 P) → (B +P 𝐶) P)
4342adantr 261 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (B +P 𝐶) P)
44 mulclpr 6553 . . . . . 6 ((A P (B +P 𝐶) P) → (A ·P (B +P 𝐶)) P)
4540, 43, 44syl2anc 391 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (A ·P (B +P 𝐶)) P)
46 distrnqg 6371 . . . . . . 7 ((f Q y Q z Q) → (f ·Q (y +Q z)) = ((f ·Q y) +Q (f ·Q z)))
4712, 18, 33, 46syl3anc 1134 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q (y +Q z)) = ((f ·Q y) +Q (f ·Q z)))
48 simprrl 491 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → f (1stA))
49 df-iplp 6451 . . . . . . . . . 10 +P = (u P, v P ↦ ⟨{w Qg Q Q (g (1stu) (1stv) w = (g +Q ))}, {w Qg Q Q (g (2ndu) (2ndv) w = (g +Q ))}⟩)
50 addclnq 6359 . . . . . . . . . 10 ((g Q Q) → (g +Q ) Q)
5149, 50genpprecll 6497 . . . . . . . . 9 ((B P 𝐶 P) → ((y (1stB) z (1st𝐶)) → (y +Q z) (1st ‘(B +P 𝐶))))
5251imp 115 . . . . . . . 8 (((B P 𝐶 P) (y (1stB) z (1st𝐶))) → (y +Q z) (1st ‘(B +P 𝐶)))
5313, 28, 14, 29, 52syl22anc 1135 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (y +Q z) (1st ‘(B +P 𝐶)))
54 df-imp 6452 . . . . . . . . 9 ·P = (u P, v P ↦ ⟨{w Qg Q Q (g (1stu) (1stv) w = (g ·Q ))}, {w Qg Q Q (g (2ndu) (2ndv) w = (g ·Q ))}⟩)
55 mulclnq 6360 . . . . . . . . 9 ((g Q Q) → (g ·Q ) Q)
5654, 55genpprecll 6497 . . . . . . . 8 ((A P (B +P 𝐶) P) → ((f (1stA) (y +Q z) (1st ‘(B +P 𝐶))) → (f ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶)))))
5756imp 115 . . . . . . 7 (((A P (B +P 𝐶) P) (f (1stA) (y +Q z) (1st ‘(B +P 𝐶)))) → (f ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
5840, 43, 48, 53, 57syl22anc 1135 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
5947, 58eqeltrrd 2112 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((f ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
60 prop 6458 . . . . . 6 ((A ·P (B +P 𝐶)) P → ⟨(1st ‘(A ·P (B +P 𝐶))), (2nd ‘(A ·P (B +P 𝐶)))⟩ P)
61 prcdnql 6467 . . . . . 6 ((⟨(1st ‘(A ·P (B +P 𝐶))), (2nd ‘(A ·P (B +P 𝐶)))⟩ P ((f ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
6260, 61sylan 267 . . . . 5 (((A ·P (B +P 𝐶)) P ((f ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
6345, 59, 62syl2anc 391 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
6439, 63sylbid 139 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x <Q f → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
652, 12, 8, 33, 20caovord2d 5612 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f <Q x ↔ (f ·Q z) <Q (x ·Q z)))
66 mulclnq 6360 . . . . . . 7 ((x Q z Q) → (x ·Q z) Q)
678, 33, 66syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q z) Q)
68 ltanqg 6384 . . . . . 6 (((f ·Q z) Q (x ·Q z) Q (x ·Q y) Q) → ((f ·Q z) <Q (x ·Q z) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
6935, 67, 25, 68syl3anc 1134 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((f ·Q z) <Q (x ·Q z) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
7065, 69bitrd 177 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f <Q x ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
71 distrnqg 6371 . . . . . . 7 ((x Q y Q z Q) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (x ·Q z)))
728, 18, 33, 71syl3anc 1134 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (x ·Q z)))
73 simprll 489 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → x (1stA))
7454, 55genpprecll 6497 . . . . . . . 8 ((A P (B +P 𝐶) P) → ((x (1stA) (y +Q z) (1st ‘(B +P 𝐶))) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶)))))
7574imp 115 . . . . . . 7 (((A P (B +P 𝐶) P) (x (1stA) (y +Q z) (1st ‘(B +P 𝐶)))) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
7640, 43, 73, 53, 75syl22anc 1135 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
7772, 76eqeltrrd 2112 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) +Q (x ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
78 prcdnql 6467 . . . . . 6 ((⟨(1st ‘(A ·P (B +P 𝐶))), (2nd ‘(A ·P (B +P 𝐶)))⟩ P ((x ·Q y) +Q (x ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
7960, 78sylan 267 . . . . 5 (((A ·P (B +P 𝐶)) P ((x ·Q y) +Q (x ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
8045, 77, 79syl2anc 391 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
8170, 80sylbid 139 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f <Q x → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
8264, 81jaod 636 . 2 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x <Q f f <Q x) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
83 ltsonq 6382 . . . . 5 <Q Or Q
84 nqtri3or 6380 . . . . 5 ((x Q f Q) → (x <Q f x = f f <Q x))
8583, 84sotritrieq 4053 . . . 4 ((x Q f Q) → (x = f ↔ ¬ (x <Q f f <Q x)))
868, 12, 85syl2anc 391 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x = f ↔ ¬ (x <Q f f <Q x)))
87 oveq1 5462 . . . . . . 7 (x = f → (x ·Q z) = (f ·Q z))
8887oveq2d 5471 . . . . . 6 (x = f → ((x ·Q y) +Q (x ·Q z)) = ((x ·Q y) +Q (f ·Q z)))
8972, 88sylan9eq 2089 . . . . 5 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) x = f) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (f ·Q z)))
9076adantr 261 . . . . 5 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) x = f) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
9189, 90eqeltrrd 2112 . . . 4 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) x = f) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
9291ex 108 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x = f → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
9386, 92sylbird 159 . 2 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (¬ (x <Q f f <Q x) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
94 ltdcnq 6381 . . . . 5 ((x Q f Q) → DECID x <Q f)
95 ltdcnq 6381 . . . . . 6 ((f Q x Q) → DECID f <Q x)
9695ancoms 255 . . . . 5 ((x Q f Q) → DECID f <Q x)
97 dcor 842 . . . . 5 (DECID x <Q f → (DECID f <Q xDECID (x <Q f f <Q x)))
9894, 96, 97sylc 56 . . . 4 ((x Q f Q) → DECID (x <Q f f <Q x))
998, 12, 98syl2anc 391 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → DECID (x <Q f f <Q x))
100 df-dc 742 . . 3 (DECID (x <Q f f <Q x) ↔ ((x <Q f f <Q x) ¬ (x <Q f f <Q x)))
10199, 100sylib 127 . 2 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x <Q f f <Q x) ¬ (x <Q f f <Q x)))
10282, 93, 101mpjaod 637 1 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  DECID wdc 741   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   ·Q cmq 6267
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