Theorem pitonnlem1 8213

Description: Lemma for pitonn 8216. 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 8188	. 2 ⊢ 1 = ⟨1R, 0R⟩
2		df-1r 8100	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3		df-i1p 7835	. . . . . . . 8 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4		df-1nqqs 7719	. . . . . . . . . . 11 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
5	4	breq2i 4138	. . . . . . . . . 10 ⊢ (𝑙 <Q 1Q ↔ 𝑙 <Q [⟨1o, 1o⟩] ~Q )
6	5	abbii 2354	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q 1Q} = {𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }
7	4	breq1i 4137	. . . . . . . . . 10 ⊢ (1Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢)
8	7	abbii 2354	. . . . . . . . 9 ⊢ {𝑢 ∣ 1Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}
9	6, 8	opeq12i 3909	. . . . . . . 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 2259	. . . . . . 7 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩
11	10	oveq1i 6095	. . . . . 6 ⊢ (1P +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)
12	11	opeq1i 3907	. . . . 5 ⊢ ⟨(1P +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩
13		eceq1 6842	. . . . 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 2259	. . 3 ⊢ 1R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
16	15	opeq1i 3907	. 2 ⊢ ⟨1R, 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩
17	1, 16	eqtr2i 2260	1 ⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1

This proof depends on syntax axioms:   = wceq 1402  {cab 2224  ⟨cop 3712   class class class wbr 4130  (class class class)co 6085  1_oc1o 6680  [cec 6805   ~_Q ceq 7647  1_Qc1q 7649   <_Q cltq 7653  1_Pc1p 7660   +_P cpp 7661   ~_R cer 7664  0_Rc0r 7666  1_Rc1r 7667  1c1 8181

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fv 5385  df-ov 6088  df-ec 6809  df-1nqqs 7719  df-i1p 7835  df-1r 8100  df-1 8188

This theorem is used by:  pitonn  8216