Theorem addcomprg 6554

Description: Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)

Assertion

Ref	Expression
addcomprg	⊢ ((A ∈ P ∧ B ∈ P) → (A +P B) = (B +P A))

Proof of Theorem addcomprg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 6458	. . . . . . . . 9 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
2		elprnql 6464	. . . . . . . . 9 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ y ∈ (1st ‘B)) → y ∈ Q)
3	1, 2	sylan 267	. . . . . . . 8 ⊢ ((B ∈ P ∧ y ∈ (1st ‘B)) → y ∈ Q)
4		prop 6458	. . . . . . . . . . . . 13 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
5		elprnql 6464	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ z ∈ (1st ‘A)) → z ∈ Q)
6	4, 5	sylan 267	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ z ∈ (1st ‘A)) → z ∈ Q)
7		addcomnqg 6365	. . . . . . . . . . . . 13 ⊢ ((y ∈ Q ∧ z ∈ Q) → (y +Q z) = (z +Q y))
8	7	eqeq2d 2048	. . . . . . . . . . . 12 ⊢ ((y ∈ Q ∧ z ∈ Q) → (x = (y +Q z) ↔ x = (z +Q y)))
9	6, 8	sylan2 270	. . . . . . . . . . 11 ⊢ ((y ∈ Q ∧ (A ∈ P ∧ z ∈ (1st ‘A))) → (x = (y +Q z) ↔ x = (z +Q y)))
10	9	anassrs 380	. . . . . . . . . 10 ⊢ (((y ∈ Q ∧ A ∈ P) ∧ z ∈ (1st ‘A)) → (x = (y +Q z) ↔ x = (z +Q y)))
11	10	rexbidva 2317	. . . . . . . . 9 ⊢ ((y ∈ Q ∧ A ∈ P) → (∃z ∈ (1st ‘A)x = (y +Q z) ↔ ∃z ∈ (1st ‘A)x = (z +Q y)))
12	11	ancoms 255	. . . . . . . 8 ⊢ ((A ∈ P ∧ y ∈ Q) → (∃z ∈ (1st ‘A)x = (y +Q z) ↔ ∃z ∈ (1st ‘A)x = (z +Q y)))
13	3, 12	sylan2 270	. . . . . . 7 ⊢ ((A ∈ P ∧ (B ∈ P ∧ y ∈ (1st ‘B))) → (∃z ∈ (1st ‘A)x = (y +Q z) ↔ ∃z ∈ (1st ‘A)x = (z +Q y)))
14	13	anassrs 380	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ y ∈ (1st ‘B)) → (∃z ∈ (1st ‘A)x = (y +Q z) ↔ ∃z ∈ (1st ‘A)x = (z +Q y)))
15	14	rexbidva 2317	. . . . 5 ⊢ ((A ∈ P ∧ B ∈ P) → (∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (y +Q z) ↔ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (z +Q y)))
16		rexcom 2468	. . . . 5 ⊢ (∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (z +Q y) ↔ ∃z ∈ (1st ‘A)∃y ∈ (1st ‘B)x = (z +Q y))
17	15, 16	syl6bb 185	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → (∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (y +Q z) ↔ ∃z ∈ (1st ‘A)∃y ∈ (1st ‘B)x = (z +Q y)))
18	17	rabbidv 2543	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → {x ∈ Q ∣ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (y +Q z)} = {x ∈ Q ∣ ∃z ∈ (1st ‘A)∃y ∈ (1st ‘B)x = (z +Q y)})
19		elprnqu 6465	. . . . . . . . 9 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ y ∈ (2nd ‘B)) → y ∈ Q)
20	1, 19	sylan 267	. . . . . . . 8 ⊢ ((B ∈ P ∧ y ∈ (2nd ‘B)) → y ∈ Q)
21		elprnqu 6465	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ z ∈ (2nd ‘A)) → z ∈ Q)
22	4, 21	sylan 267	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ z ∈ (2nd ‘A)) → z ∈ Q)
23	22, 8	sylan2 270	. . . . . . . . . . 11 ⊢ ((y ∈ Q ∧ (A ∈ P ∧ z ∈ (2nd ‘A))) → (x = (y +Q z) ↔ x = (z +Q y)))
24	23	anassrs 380	. . . . . . . . . 10 ⊢ (((y ∈ Q ∧ A ∈ P) ∧ z ∈ (2nd ‘A)) → (x = (y +Q z) ↔ x = (z +Q y)))
25	24	rexbidva 2317	. . . . . . . . 9 ⊢ ((y ∈ Q ∧ A ∈ P) → (∃z ∈ (2nd ‘A)x = (y +Q z) ↔ ∃z ∈ (2nd ‘A)x = (z +Q y)))
26	25	ancoms 255	. . . . . . . 8 ⊢ ((A ∈ P ∧ y ∈ Q) → (∃z ∈ (2nd ‘A)x = (y +Q z) ↔ ∃z ∈ (2nd ‘A)x = (z +Q y)))
27	20, 26	sylan2 270	. . . . . . 7 ⊢ ((A ∈ P ∧ (B ∈ P ∧ y ∈ (2nd ‘B))) → (∃z ∈ (2nd ‘A)x = (y +Q z) ↔ ∃z ∈ (2nd ‘A)x = (z +Q y)))
28	27	anassrs 380	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ y ∈ (2nd ‘B)) → (∃z ∈ (2nd ‘A)x = (y +Q z) ↔ ∃z ∈ (2nd ‘A)x = (z +Q y)))
29	28	rexbidva 2317	. . . . 5 ⊢ ((A ∈ P ∧ B ∈ P) → (∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (y +Q z) ↔ ∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (z +Q y)))
30		rexcom 2468	. . . . 5 ⊢ (∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (z +Q y) ↔ ∃z ∈ (2nd ‘A)∃y ∈ (2nd ‘B)x = (z +Q y))
31	29, 30	syl6bb 185	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → (∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (y +Q z) ↔ ∃z ∈ (2nd ‘A)∃y ∈ (2nd ‘B)x = (z +Q y)))
32	31	rabbidv 2543	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → {x ∈ Q ∣ ∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (y +Q z)} = {x ∈ Q ∣ ∃z ∈ (2nd ‘A)∃y ∈ (2nd ‘B)x = (z +Q y)})
33	18, 32	opeq12d 3548	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → ⟨{x ∈ Q ∣ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (y +Q z)}, {x ∈ Q ∣ ∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (y +Q z)}⟩ = ⟨{x ∈ Q ∣ ∃z ∈ (1st ‘A)∃y ∈ (1st ‘B)x = (z +Q y)}, {x ∈ Q ∣ ∃z ∈ (2nd ‘A)∃y ∈ (2nd ‘B)x = (z +Q y)}⟩)
34		plpvlu 6521	. . 3 ⊢ ((B ∈ P ∧ A ∈ P) → (B +P A) = ⟨{x ∈ Q ∣ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (y +Q z)}, {x ∈ Q ∣ ∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (y +Q z)}⟩)
35	34	ancoms 255	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (B +P A) = ⟨{x ∈ Q ∣ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘A)x = (y +Q z)}, {x ∈ Q ∣ ∃y ∈ (2nd ‘B)∃z ∈ (2nd ‘A)x = (y +Q z)}⟩)
36		plpvlu 6521	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (A +P B) = ⟨{x ∈ Q ∣ ∃z ∈ (1st ‘A)∃y ∈ (1st ‘B)x = (z +Q y)}, {x ∈ Q ∣ ∃z ∈ (2nd ‘A)∃y ∈ (2nd ‘B)x = (z +Q y)}⟩)
37	33, 35, 36	3eqtr4rd 2080	1 ⊢ ((A ∈ P ∧ B ∈ P) → (A +P B) = (B +P A))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  {crab 2304  ⟨cop 3370  ‘cfv 4845  (class class class)co 5455  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264   +_Q cplq 6266  Pcnp 6275   +_P cpp 6277

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-plpq 6328  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-inp 6449  df-iplp 6451

This theorem is referenced by:  addextpr  6593  cauappcvgprlemcan  6616  enrer  6663  addcmpblnr  6667  mulcmpblnrlemg  6668  ltsrprg  6675  addcomsrg  6683  mulcomsrg  6685  mulasssrg  6686  distrsrg  6687  lttrsr  6690  ltposr  6691  ltsosr  6692  0lt1sr  6693  0idsr  6695  1idsr  6696  ltasrg  6698  recexgt0sr  6701  mulgt0sr  6704  aptisr  6705  mulextsr1lem  6706  archsr  6708  pitonnlem1p1  6742