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Theorem pitonnlem1 7843
Description: Lemma for pitonn 7846. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
Assertion
Ref Expression
pitonnlem1 ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
Distinct variable group:   𝑢,𝑙

Proof of Theorem pitonnlem1
StepHypRef Expression
1 df-1 7818 . 2 1 = ⟨1R, 0R
2 df-1r 7730 . . . 4 1R = [⟨(1P +P 1P), 1P⟩] ~R
3 df-i1p 7465 . . . . . . . 8 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4 df-1nqqs 7349 . . . . . . . . . . 11 1Q = [⟨1o, 1o⟩] ~Q
54breq2i 4011 . . . . . . . . . 10 (𝑙 <Q 1Q𝑙 <Q [⟨1o, 1o⟩] ~Q )
65abbii 2293 . . . . . . . . 9 {𝑙𝑙 <Q 1Q} = {𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }
74breq1i 4010 . . . . . . . . . 10 (1Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢)
87abbii 2293 . . . . . . . . 9 {𝑢 ∣ 1Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}
96, 8opeq12i 3783 . . . . . . . 8 ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩
103, 9eqtri 2198 . . . . . . 7 1P = ⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩
1110oveq1i 5884 . . . . . 6 (1P +P 1P) = (⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)
1211opeq1i 3781 . . . . 5 ⟨(1P +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P
13 eceq1 6569 . . . . 5 (⟨(1P +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ → [⟨(1P +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
1412, 13ax-mp 5 . . . 4 [⟨(1P +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
152, 14eqtri 2198 . . 3 1R = [⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
1615opeq1i 3781 . 2 ⟨1R, 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R
171, 16eqtr2i 2199 1 ⟨[⟨(⟨{𝑙𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
Colors of variables: wff set class
Syntax hints:   = wceq 1353  {cab 2163  cop 3595   class class class wbr 4003  (class class class)co 5874  1oc1o 6409  [cec 6532   ~Q ceq 7277  1Qc1q 7279   <Q cltq 7283  1Pc1p 7290   +P cpp 7291   ~R cer 7294  0Rc0r 7296  1Rc1r 7297  1c1 7811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fv 5224  df-ov 5877  df-ec 6536  df-1nqqs 7349  df-i1p 7465  df-1r 7730  df-1 7818
This theorem is referenced by:  pitonn  7846
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