Theorem pitonnlem1p1 8033

Description: Lemma for pitonn 8035. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)

Assertion

Ref	Expression
pitonnlem1p1	⊢ (𝐴 ∈ P → [⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R )

Proof of Theorem pitonnlem1p1

Step	Hyp	Ref	Expression
1		1pr 7741	. . . . . 6 ⊢ 1P ∈ P
2		addclpr 7724	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
3	1, 1, 2	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
4		addcomprg 7765	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
5	3, 4	mpan2 425	. . . 4 ⊢ (𝐴 ∈ P → (𝐴 +P (1P +P 1P)) = ((1P +P 1P) +P 𝐴))
6	5	oveq1d 6016	. . 3 ⊢ (𝐴 ∈ P → ((𝐴 +P (1P +P 1P)) +P 1P) = (((1P +P 1P) +P 𝐴) +P 1P))
7		addassprg 7766	. . . 4 ⊢ (((1P +P 1P) ∈ P ∧ 𝐴 ∈ P ∧ 1P ∈ P) → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
8	3, 1, 7	mp3an13 1362	. . 3 ⊢ (𝐴 ∈ P → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
9	6, 8	eqtrd 2262	. 2 ⊢ (𝐴 ∈ P → ((𝐴 +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
10		addclpr 7724	. . . 4 ⊢ ((𝐴 ∈ P ∧ (1P +P 1P) ∈ P) → (𝐴 +P (1P +P 1P)) ∈ P)
11	3, 10	mpan2 425	. . 3 ⊢ (𝐴 ∈ P → (𝐴 +P (1P +P 1P)) ∈ P)
12	3	a1i 9	. . 3 ⊢ (𝐴 ∈ P → (1P +P 1P) ∈ P)
13		addclpr 7724	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P)
14	1, 13	mpan2 425	. . 3 ⊢ (𝐴 ∈ P → (𝐴 +P 1P) ∈ P)
15	1	a1i 9	. . 3 ⊢ (𝐴 ∈ P → 1P ∈ P)
16		enreceq 7923	. . 3 ⊢ ((((𝐴 +P (1P +P 1P)) ∈ P ∧ (1P +P 1P) ∈ P) ∧ ((𝐴 +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R ↔ ((𝐴 +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P))))
17	11, 12, 14, 15, 16	syl22anc 1272	. 2 ⊢ (𝐴 ∈ P → ([⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R ↔ ((𝐴 +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P))))
18	9, 17	mpbird 167	1 ⊢ (𝐴 ∈ P → [⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R )

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ⟨cop 3669  (class class class)co 6001  [cec 6678  Pcnp 7478  1_Pc1p 7479   +_P cpp 7480   ~_R cer 7483

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-2o 6563  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-enq0 7611  df-nq0 7612  df-0nq0 7613  df-plq0 7614  df-mq0 7615  df-inp 7653  df-i1p 7654  df-iplp 7655  df-enr 7913

This theorem is referenced by:  pitonnlem2  8034