Theorem pitonnlem1 7940

Description: Lemma for pitonn 7943. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)

Assertion

Ref	Expression
pitonnlem1	⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <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

Step	Hyp	Ref	Expression
1		df-1 7915	. 2 ⊢ 1 = ⟨1R, 0R⟩
2		df-1r 7827	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3		df-i1p 7562	. . . . . . . 8 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4		df-1nqqs 7446	. . . . . . . . . . 11 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
5	4	breq2i 4051	. . . . . . . . . 10 ⊢ (𝑙 <Q 1Q ↔ 𝑙 <Q [⟨1o, 1o⟩] ~Q )
6	5	abbii 2320	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q 1Q} = {𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }
7	4	breq1i 4050	. . . . . . . . . 10 ⊢ (1Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢)
8	7	abbii 2320	. . . . . . . . 9 ⊢ {𝑢 ∣ 1Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}
9	6, 8	opeq12i 3823	. . . . . . . 8 ⊢ ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩
10	3, 9	eqtri 2225	. . . . . . 7 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩
11	10	oveq1i 5944	. . . . . 6 ⊢ (1P +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)
12	11	opeq1i 3821	. . . . 5 ⊢ ⟨(1P +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩
13		eceq1 6645	. . . . 5 ⊢ (⟨(1P +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ → [⟨(1P +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
14	12, 13	ax-mp 5	. . . 4 ⊢ [⟨(1P +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
15	2, 14	eqtri 2225	. . 3 ⊢ 1R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
16	15	opeq1i 3821	. 2 ⊢ ⟨1R, 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩
17	1, 16	eqtr2i 2226	1 ⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1

Syntax hints:   = wceq 1372  {cab 2190  ⟨cop 3635   class class class wbr 4043  (class class class)co 5934  1_oc1o 6485  [cec 6608   ~_Q ceq 7374  1_Qc1q 7376   <_Q cltq 7380  1_Pc1p 7387   +_P cpp 7388   ~_R cer 7391  0_Rc0r 7393  1_Rc1r 7394  1c1 7908

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-xp 4679  df-cnv 4681  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fv 5276  df-ov 5937  df-ec 6612  df-1nqqs 7446  df-i1p 7562  df-1r 7827  df-1 7915

This theorem is referenced by:  pitonn  7943