Theorem prsradd 7981

Description: Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)

Assertion

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

Proof of Theorem prsradd

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1pr 7749	. . . 4 ⊢ 1P ∈ P
2		addclpr 7732	. . . 4 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P)
3	1, 2	mpan2 425	. . 3 ⊢ (𝐴 ∈ P → (𝐴 +P 1P) ∈ P)
4		addclpr 7732	. . . 4 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 +P 1P) ∈ P)
5	1, 4	mpan2 425	. . 3 ⊢ (𝐵 ∈ P → (𝐵 +P 1P) ∈ P)
6		addsrpr 7940	. . . . 5 ⊢ ((((𝐴 +P 1P) ∈ P ∧ 1P ∈ P) ∧ ((𝐵 +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
7	1, 6	mpanl2 435	. . . 4 ⊢ (((𝐴 +P 1P) ∈ P ∧ ((𝐵 +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
8	1, 7	mpanr2 438	. . 3 ⊢ (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
9	3, 5, 8	syl2an 289	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
10		simpl 109	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴 ∈ P)
11	1	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 1P ∈ P)
12		simpr 110	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐵 ∈ P)
13		addcomprg 7773	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
14	13	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
15		addassprg 7774	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ)))
16	15	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ)))
17		addclpr 7732	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P)
18	17	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) ∈ P)
19	10, 11, 12, 14, 16, 11, 18	caov4d 6196	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((𝐴 +P 𝐵) +P (1P +P 1P)))
20		addclpr 7732	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
21		addclpr 7732	. . . . . . . . 9 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
22	1, 1, 21	mp2an 426	. . . . . . . 8 ⊢ (1P +P 1P) ∈ P
23	22	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (1P +P 1P) ∈ P)
24		addcomprg 7773	. . . . . . 7 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (1P +P 1P) ∈ P) → ((𝐴 +P 𝐵) +P (1P +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
25	20, 23, 24	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) +P (1P +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
26	19, 25	eqtrd 2262	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
27	26	oveq1d 6022	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P))
28		addassprg 7774	. . . . 5 ⊢ (((1P +P 1P) ∈ P ∧ (𝐴 +P 𝐵) ∈ P ∧ 1P ∈ P) → (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
29	23, 20, 11, 28	syl3anc 1271	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
30	27, 29	eqtrd 2262	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
31		addclpr 7732	. . . . 5 ⊢ (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
32	3, 5, 31	syl2an 289	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
33		addclpr 7732	. . . . 5 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ 1P ∈ P) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
34	20, 11, 33	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
35		enreceq 7931	. . . . . 6 ⊢ (((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ (1P +P 1P) ∈ P) ∧ (((𝐴 +P 𝐵) +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
36	1, 35	mpanr2 438	. . . . 5 ⊢ (((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ (1P +P 1P) ∈ P) ∧ ((𝐴 +P 𝐵) +P 1P) ∈ P) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
37	22, 36	mpanl2 435	. . . 4 ⊢ ((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ ((𝐴 +P 𝐵) +P 1P) ∈ P) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
38	32, 34, 37	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
39	30, 38	mpbird 167	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R )
40	9, 39	eqtr2d 2263	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R = ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ⟨cop 3669  (class class class)co 6007  [cec 6686  Pcnp 7486  1_Pc1p 7487   +_P cpp 7488   ~_R cer 7491   +_R cplr 7496

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 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-lti 7502  df-plpq 7539  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-mqqs 7545  df-1nqqs 7546  df-rq 7547  df-ltnqqs 7548  df-enq0 7619  df-nq0 7620  df-0nq0 7621  df-plq0 7622  df-mq0 7623  df-inp 7661  df-i1p 7662  df-iplp 7663  df-enr 7921  df-nr 7922  df-plr 7923

This theorem is referenced by:  caucvgsrlemcau  7988  caucvgsrlemgt1  7990