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Theorem 1idpr 9836
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 2915 . . . . 5 (∃𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑔(𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔)))
2 19.42v 1916 . . . . . 6 (∃𝑔(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)))
3 elprnq 9798 . . . . . . . . . 10 ((𝐴P𝑓𝐴) → 𝑓Q)
4 breq1 4647 . . . . . . . . . . 11 (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓 ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
5 df-1p 9789 . . . . . . . . . . . . 13 1P = {𝑔𝑔 <Q 1Q}
65abeq2i 2733 . . . . . . . . . . . 12 (𝑔 ∈ 1P𝑔 <Q 1Q)
7 ltmnq 9779 . . . . . . . . . . . . 13 (𝑓Q → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q)))
8 mulidnq 9770 . . . . . . . . . . . . . 14 (𝑓Q → (𝑓 ·Q 1Q) = 𝑓)
98breq2d 4656 . . . . . . . . . . . . 13 (𝑓Q → ((𝑓 ·Q 𝑔) <Q (𝑓 ·Q 1Q) ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
107, 9bitrd 268 . . . . . . . . . . . 12 (𝑓Q → (𝑔 <Q 1Q ↔ (𝑓 ·Q 𝑔) <Q 𝑓))
116, 10syl5rbb 273 . . . . . . . . . . 11 (𝑓Q → ((𝑓 ·Q 𝑔) <Q 𝑓𝑔 ∈ 1P))
124, 11sylan9bbr 736 . . . . . . . . . 10 ((𝑓Q𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ 1P))
133, 12sylan 488 . . . . . . . . 9 (((𝐴P𝑓𝐴) ∧ 𝑥 = (𝑓 ·Q 𝑔)) → (𝑥 <Q 𝑓𝑔 ∈ 1P))
1413ex 450 . . . . . . . 8 ((𝐴P𝑓𝐴) → (𝑥 = (𝑓 ·Q 𝑔) → (𝑥 <Q 𝑓𝑔 ∈ 1P)))
1514pm5.32rd 671 . . . . . . 7 ((𝐴P𝑓𝐴) → ((𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔))))
1615exbidv 1848 . . . . . 6 ((𝐴P𝑓𝐴) → (∃𝑔(𝑥 <Q 𝑓𝑥 = (𝑓 ·Q 𝑔)) ↔ ∃𝑔(𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔))))
172, 16syl5rbbr 275 . . . . 5 ((𝐴P𝑓𝐴) → (∃𝑔(𝑔 ∈ 1P𝑥 = (𝑓 ·Q 𝑔)) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
181, 17syl5bb 272 . . . 4 ((𝐴P𝑓𝐴) → (∃𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔) ↔ (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
1918rexbidva 3045 . . 3 (𝐴P → (∃𝑓𝐴𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔) ↔ ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
20 1pr 9822 . . . 4 1PP
21 df-mp 9791 . . . . 5 ·P = (𝑦P, 𝑧P ↦ {𝑤 ∣ ∃𝑢𝑦𝑣𝑧 𝑤 = (𝑢 ·Q 𝑣)})
22 mulclnq 9754 . . . . 5 ((𝑢Q𝑣Q) → (𝑢 ·Q 𝑣) ∈ Q)
2321, 22genpelv 9807 . . . 4 ((𝐴P ∧ 1PP) → (𝑥 ∈ (𝐴 ·P 1P) ↔ ∃𝑓𝐴𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔)))
2420, 23mpan2 706 . . 3 (𝐴P → (𝑥 ∈ (𝐴 ·P 1P) ↔ ∃𝑓𝐴𝑔 ∈ 1P 𝑥 = (𝑓 ·Q 𝑔)))
25 prnmax 9802 . . . . . 6 ((𝐴P𝑥𝐴) → ∃𝑓𝐴 𝑥 <Q 𝑓)
26 ltrelnq 9733 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
2726brel 5158 . . . . . . . . . 10 (𝑥 <Q 𝑓 → (𝑥Q𝑓Q))
28 vex 3198 . . . . . . . . . . . . . 14 𝑓 ∈ V
29 vex 3198 . . . . . . . . . . . . . 14 𝑥 ∈ V
30 fvex 6188 . . . . . . . . . . . . . 14 (*Q𝑓) ∈ V
31 mulcomnq 9760 . . . . . . . . . . . . . 14 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
32 mulassnq 9766 . . . . . . . . . . . . . 14 ((𝑦 ·Q 𝑧) ·Q 𝑤) = (𝑦 ·Q (𝑧 ·Q 𝑤))
3328, 29, 30, 31, 32caov12 6847 . . . . . . . . . . . . 13 (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q (𝑓 ·Q (*Q𝑓)))
34 recidnq 9772 . . . . . . . . . . . . . 14 (𝑓Q → (𝑓 ·Q (*Q𝑓)) = 1Q)
3534oveq2d 6651 . . . . . . . . . . . . 13 (𝑓Q → (𝑥 ·Q (𝑓 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
3633, 35syl5eq 2666 . . . . . . . . . . . 12 (𝑓Q → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = (𝑥 ·Q 1Q))
37 mulidnq 9770 . . . . . . . . . . . 12 (𝑥Q → (𝑥 ·Q 1Q) = 𝑥)
3836, 37sylan9eqr 2676 . . . . . . . . . . 11 ((𝑥Q𝑓Q) → (𝑓 ·Q (𝑥 ·Q (*Q𝑓))) = 𝑥)
3938eqcomd 2626 . . . . . . . . . 10 ((𝑥Q𝑓Q) → 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
40 ovex 6663 . . . . . . . . . . 11 (𝑥 ·Q (*Q𝑓)) ∈ V
41 oveq2 6643 . . . . . . . . . . . 12 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑓 ·Q 𝑔) = (𝑓 ·Q (𝑥 ·Q (*Q𝑓))))
4241eqeq2d 2630 . . . . . . . . . . 11 (𝑔 = (𝑥 ·Q (*Q𝑓)) → (𝑥 = (𝑓 ·Q 𝑔) ↔ 𝑥 = (𝑓 ·Q (𝑥 ·Q (*Q𝑓)))))
4340, 42spcev 3295 . . . . . . . . . 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 575 . . . . . . 7 (𝑓𝐴 → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
4746reximia 3006 . . . . . 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 450 . . . 4 (𝐴P → (𝑥𝐴 → ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
50 prcdnq 9800 . . . . . 6 ((𝐴P𝑓𝐴) → (𝑥 <Q 𝑓𝑥𝐴))
5150adantrd 484 . . . . 5 ((𝐴P𝑓𝐴) → ((𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)) → 𝑥𝐴))
5251rexlimdva 3027 . . . 4 (𝐴P → (∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔)) → 𝑥𝐴))
5349, 52impbid 202 . . 3 (𝐴P → (𝑥𝐴 ↔ ∃𝑓𝐴 (𝑥 <Q 𝑓 ∧ ∃𝑔 𝑥 = (𝑓 ·Q 𝑔))))
5419, 24, 533bitr4d 300 . 2 (𝐴P → (𝑥 ∈ (𝐴 ·P 1P) ↔ 𝑥𝐴))
5554eqrdv 2618 1 (𝐴P → (𝐴 ·P 1P) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  wrex 2910   class class class wbr 4644  cfv 5876  (class class class)co 6635  Qcnq 9659  1Qc1q 9660   ·Q cmq 9663  *Qcrq 9664   <Q cltq 9665  Pcnp 9666  1Pc1p 9667   ·P cmp 9669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-ni 9679  df-pli 9680  df-mi 9681  df-lti 9682  df-plpq 9715  df-mpq 9716  df-ltpq 9717  df-enq 9718  df-nq 9719  df-erq 9720  df-plq 9721  df-mq 9722  df-1nq 9723  df-rq 9724  df-ltnq 9725  df-np 9788  df-1p 9789  df-mp 9791
This theorem is referenced by:  m1m1sr  9899  1idsr  9904
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