Theorem pitonnlem1p1 7847

Description: Lemma for pitonn 7849. 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 7555	. . . . . 6 ⊢ 1P ∈ P
2		addclpr 7538	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
3	1, 1, 2	mp2an 426	. . . . 5 ⊢ (1P +P 1P) ∈ P
4		addcomprg 7579	. . . . 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 5892	. . 3 ⊢ (𝐴 ∈ P → ((𝐴 +P (1P +P 1P)) +P 1P) = (((1P +P 1P) +P 𝐴) +P 1P))
7		addassprg 7580	. . . 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 1328	. . 3 ⊢ (𝐴 ∈ P → (((1P +P 1P) +P 𝐴) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
9	6, 8	eqtrd 2210	. 2 ⊢ (𝐴 ∈ P → ((𝐴 +P (1P +P 1P)) +P 1P) = ((1P +P 1P) +P (𝐴 +P 1P)))
10		addclpr 7538	. . . 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 7538	. . . 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 7737	. . 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 1239	. 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 1353   ∈ wcel 2148  ⟨cop 3597  (class class class)co 5877  [cec 6535  Pcnp 7292  1_Pc1p 7293   +_P cpp 7294   ~_R cer 7297

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 614  ax-in2 615  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-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-enr 7727

This theorem is referenced by:  pitonnlem2  7848