Theorem pitonnlem1 7846

Description: Lemma for pitonn 7849. 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 7821	. 2 ⊢ 1 = ⟨1R, 0R⟩
2		df-1r 7733	. . . 4 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
3		df-i1p 7468	. . . . . . . 8 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
4		df-1nqqs 7352	. . . . . . . . . . 11 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
5	4	breq2i 4013	. . . . . . . . . 10 ⊢ (𝑙 <Q 1Q ↔ 𝑙 <Q [⟨1o, 1o⟩] ~Q )
6	5	abbii 2293	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q 1Q} = {𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }
7	4	breq1i 4012	. . . . . . . . . 10 ⊢ (1Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢)
8	7	abbii 2293	. . . . . . . . 9 ⊢ {𝑢 ∣ 1Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}
9	6, 8	opeq12i 3785	. . . . . . . 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 2198	. . . . . . 7 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩
11	10	oveq1i 5887	. . . . . 6 ⊢ (1P +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)
12	11	opeq1i 3783	. . . . 5 ⊢ ⟨(1P +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩
13		eceq1 6572	. . . . 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 2198	. . 3 ⊢ 1R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
16	15	opeq1i 3783	. 2 ⊢ ⟨1R, 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩
17	1, 16	eqtr2i 2199	1 ⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1

Syntax hints:   = wceq 1353  {cab 2163  ⟨cop 3597   class class class wbr 4005  (class class class)co 5877  1_oc1o 6412  [cec 6535   ~_Q ceq 7280  1_Qc1q 7282   <_Q cltq 7286  1_Pc1p 7293   +_P cpp 7294   ~_R cer 7297  0_Rc0r 7299  1_Rc1r 7300  1c1 7814

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 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fv 5226  df-ov 5880  df-ec 6539  df-1nqqs 7352  df-i1p 7468  df-1r 7733  df-1 7821

This theorem is referenced by:  pitonn  7849