Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  1idpr Structured version   Visualization version   GIF version

Theorem 1idpr 10440
 Description: 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
1idpr (𝐴P → (𝐴 ·P 1P) = 𝐴)

Proof of Theorem 1idpr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 3136 . . . . 5 (∃𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑔(𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔)))
2 elprnq 10402 . . . . . . . . . 10 ((𝐴P𝑓𝐴) → 𝑓Q)
3 breq1 5045 . . . . . . . . . . 11 (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
4 df-1p 10393 . . . . . . . . . . . . 13 1P = {𝑔𝑔 <Q 1Q}
54abeq2i 2949 . . . . . . . . . . . 12 (𝑔 ∈ 1P𝑔 <Q 1Q)
6 ltmnq 10383 . . . . . . . . . . . . 13 (𝑓Q → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
7 mulidnq 10374 . . . . . . . . . . . . . 14 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
87breq2d 5054 . . . . . . . . . . . . 13 (𝑓Q → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
96, 8bitrd 282 . . . . . . . . . . . 12 (𝑓Q → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
105, 9syl5rbb 287 . . . . . . . . . . 11 (𝑓Q → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ 1P))
113, 10sylan9bbr 514 . . . . . . . . . 10 ((𝑓Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ 1P))
122, 11sylan 583 . . . . . . . . 9 (((𝐴P𝑓𝐴) ∧ 𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ 1P))
1312ex 416 . . . . . . . 8 ((𝐴P𝑓𝐴) → (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓𝑔 ∈ 1P)))
1413pm5.32rd 581 . . . . . . 7 ((𝐴P𝑓𝐴) → ((𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔))))
1514exbidv 1922 . . . . . 6 ((𝐴P𝑓𝐴) → (∃𝑔(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ ∃𝑔(𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔))))
16 19.42v 1954 . . . . . 6 (∃𝑔(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)))
1715, 16bitr3di 289 . . . . 5 ((𝐴P𝑓𝐴) → (∃𝑔(𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
181, 17syl5bb 286 . . . 4 ((𝐴P𝑓𝐴) → (∃𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
1918rexbidva 3282 . . 3 (𝐴P → (∃𝑓𝐴𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
20 1pr 10426 . . . 4 1PP
21 df-mp 10395 . . . . 5 ·P = (𝑦P, 𝑧P ↦ {𝑤 ∣ ∃𝑢𝑦𝑣𝑧 𝑤 = (𝑢 ·Q 𝑣)})
22 mulclnq 10358 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
2321, 22genpelv 10411 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (𝐴 ·P 1P) ↔ ∃𝑓𝐴𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔)))
2420, 23mpan2 690 . . 3 (𝐴P → (𝑥 ∈ (𝐴 ·P 1P) ↔ ∃𝑓𝐴𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔)))
25 prnmax 10406 . . . . . 6 ((𝐴P𝑥𝐴) → ∃𝑓𝐴 𝑥 <Q 𝑓)
26 ltrelnq 10337 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
2726brel 5594 . . . . . . . . . 10 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
28 vex 3472 . . . . . . . . . . . . . 14 𝑓 ∈ V
29 vex 3472 . . . . . . . . . . . . . 14 𝑥 ∈ V
30 fvex 6665 . . . . . . . . . . . . . 14 (*Q𝑓) ∈ V
31 mulcomnq 10364 . . . . . . . . . . . . . 14 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
32 mulassnq 10370 . . . . . . . . . . . . . 14 ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤))
3328, 29, 30, 31, 32caov12 7361 . . . . . . . . . . . . 13 (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓)))
34 recidnq 10376 . . . . . . . . . . . . . 14 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
3534oveq2d 7156 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
3633, 35syl5eq 2869 . . . . . . . . . . . 12 (𝑓Q → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
37 mulidnq 10374 . . . . . . . . . . . 12 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
3836, 37sylan9eqr 2879 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = 𝑥)
3938eqcomd 2828 . . . . . . . . . 10 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
40 ovex 7173 . . . . . . . . . . 11 (𝑥 ·Q (*Q𝑓)) ∈ V
41 oveq2 7148 . . . . . . . . . . . 12 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q 𝑔) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
4241eqeq2d 2833 . . . . . . . . . . 11 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q 𝑔) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
4340, 42spcev 3582 . . . . . . . . . 10 (𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) → ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))
4427, 39, 433syl 18 . . . . . . . . 9 (𝑥 <Q 𝑓 → ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))
4544a1i 11 . . . . . . . 8 (𝑓𝐴 → (𝑥 <Q 𝑓 → ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)))
4645ancld 554 . . . . . . 7 (𝑓𝐴 → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
4746reximia 3230 . . . . . 6 (∃𝑓𝐴 𝑥 <Q 𝑓 → ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)))
4825, 47syl 17 . . . . 5 ((𝐴P𝑥𝐴) → ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)))
4948ex 416 . . . 4 (𝐴P → (𝑥𝐴 → ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
50 prcdnq 10404 . . . . . 6 ((𝐴P𝑓𝐴) → (𝑥 <Q 𝑓𝑥𝐴))
5150adantrd 495 . . . . 5 ((𝐴P𝑓𝐴) → ((𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)) → 𝑥𝐴))
5251rexlimdva 3270 . . . 4 (𝐴P → (∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)) → 𝑥𝐴))
5349, 52impbid 215 . . 3 (𝐴P → (𝑥𝐴 ↔ ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
5419, 24, 533bitr4d 314 . 2 (𝐴P → (𝑥 ∈ (𝐴 ·P 1P) ↔ 𝑥𝐴))
5554eqrdv 2820 1 (𝐴P → (𝐴 ·P 1P) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2114  ∃wrex 3131   class class class wbr 5042  ‘cfv 6334  (class class class)co 7140  Qcnq 10263  1Qc1q 10264   ·Q cmq 10267  *Qcrq 10268
 Copyright terms: Public domain W3C validator