Theorem pitonnlem1 7905

Description: Lemma for pitonn 7908. 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 7880	. 2 ⊢ 1 = ⟨1R, 0R⟩
2		df-1r 7792	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3		df-i1p 7527	. . . . . . . 8 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4		df-1nqqs 7411	. . . . . . . . . . 11 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
5	4	breq2i 4037	. . . . . . . . . 10 ⊢ (𝑙 <Q 1Q ↔ 𝑙 <Q [⟨1o, 1o⟩] ~Q )
6	5	abbii 2309	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q 1Q} = {𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }
7	4	breq1i 4036	. . . . . . . . . 10 ⊢ (1Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢)
8	7	abbii 2309	. . . . . . . . 9 ⊢ {𝑢 ∣ 1Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}
9	6, 8	opeq12i 3809	. . . . . . . 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 2214	. . . . . . 7 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩
11	10	oveq1i 5928	. . . . . 6 ⊢ (1P +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)
12	11	opeq1i 3807	. . . . 5 ⊢ ⟨(1P +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩
13		eceq1 6622	. . . . 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 2214	. . 3 ⊢ 1R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
16	15	opeq1i 3807	. 2 ⊢ ⟨1R, 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩
17	1, 16	eqtr2i 2215	1 ⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1

Syntax hints:   = wceq 1364  {cab 2179  ⟨cop 3621   class class class wbr 4029  (class class class)co 5918  1_oc1o 6462  [cec 6585   ~_Q ceq 7339  1_Qc1q 7341   <_Q cltq 7345  1_Pc1p 7352   +_P cpp 7353   ~_R cer 7356  0_Rc0r 7358  1_Rc1r 7359  1c1 7873

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fv 5262  df-ov 5921  df-ec 6589  df-1nqqs 7411  df-i1p 7527  df-1r 7792  df-1 7880

This theorem is referenced by:  pitonn  7908