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Theorem addcomprg 7386

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 7283	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7289	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
3	1, 2	sylan 281	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
4		prop 7283	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7289	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
6	4, 5	sylan 281	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
7		addcomnqg 7189	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) = (𝑧 +_Q 𝑦))
8	7	eqeq2d 2151	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
9	6, 8	sylan2 284	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴))) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
10	9	anassrs 397	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑧 ∈ (1^st ‘𝐴)) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
11	10	rexbidva 2434	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
12	11	ancoms 266	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ Q) → (∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
13	3, 12	sylan2 284	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵))) → (∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
14	13	anassrs 397	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
15	14	rexbidva 2434	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
16		rexcom 2595	. . . . 5 ⊢ (∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑧 +_Q 𝑦) ↔ ∃𝑧 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑥 = (𝑧 +_Q 𝑦))
17	15, 16	syl6bb 195	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)))
18	17	rabbidv 2675	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)} = {𝑥 ∈ Q ∣ ∃𝑧 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)})
19		elprnqu 7290	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐵)) → 𝑦 ∈ Q)
20	1, 19	sylan 281	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐵)) → 𝑦 ∈ Q)
21		elprnqu 7290	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐴)) → 𝑧 ∈ Q)
22	4, 21	sylan 281	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐴)) → 𝑧 ∈ Q)
23	22, 8	sylan2 284	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐴))) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
24	23	anassrs 397	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑧 ∈ (2^nd ‘𝐴)) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
25	24	rexbidva 2434	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
26	25	ancoms 266	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ Q) → (∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
27	20, 26	sylan2 284	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐵))) → (∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
28	27	anassrs 397	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑦 ∈ (2^nd ‘𝐵)) → (∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
29	28	rexbidva 2434	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
30		rexcom 2595	. . . . 5 ⊢ (∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 +_Q 𝑦) ↔ ∃𝑧 ∈ (2^nd ‘𝐴)∃𝑦 ∈ (2^nd ‘𝐵)𝑥 = (𝑧 +_Q 𝑦))
31	29, 30	syl6bb 195	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (2^nd ‘𝐴)∃𝑦 ∈ (2^nd ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)))
32	31	rabbidv 2675	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)} = {𝑥 ∈ Q ∣ ∃𝑧 ∈ (2^nd ‘𝐴)∃𝑦 ∈ (2^nd ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)})
33	18, 32	opeq12d 3713	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑧 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)}, {𝑥 ∈ Q ∣ ∃𝑧 ∈ (2^nd ‘𝐴)∃𝑦 ∈ (2^nd ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)}⟩)
34		plpvlu 7346	. . 3 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 +_P 𝐴) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)}⟩)
35	34	ancoms 266	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐵 +_P 𝐴) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2^nd ‘𝐵)∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧)}⟩)
36		plpvlu 7346	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑧 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)}, {𝑥 ∈ Q ∣ ∃𝑧 ∈ (2^nd ‘𝐴)∃𝑦 ∈ (2^nd ‘𝐵)𝑥 = (𝑧 +_Q 𝑦)}⟩)
37	33, 35, 36	3eqtr4rd 2183	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) = (𝐵 +_P 𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 {crab 2420 ⟨cop 3530 ‘cfv 5123 (class class class)co 5774 1^st c1st 6036 2^nd c2nd 6037 Qcnq 7088 +_Q cplq 7090 Pcnp 7099 +_P cpp 7101

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-plpq 7152 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-inp 7274 df-iplp 7276

This theorem is referenced by: prplnqu 7428 addextpr 7429 caucvgprlemcanl 7452 caucvgprprlemnkltj 7497 caucvgprprlemnbj 7501 caucvgprprlemmu 7503 caucvgprprlemloc 7511 caucvgprprlemexbt 7514 caucvgprprlemexb 7515 caucvgprprlemaddq 7516 enrer 7543 addcmpblnr 7547 mulcmpblnrlemg 7548 ltsrprg 7555 addcomsrg 7563 mulcomsrg 7565 mulasssrg 7566 distrsrg 7567 lttrsr 7570 ltposr 7571 ltsosr 7572 0lt1sr 7573 0idsr 7575 1idsr 7576 ltasrg 7578 recexgt0sr 7581 mulgt0sr 7586 aptisr 7587 mulextsr1lem 7588 archsr 7590 srpospr 7591 prsrpos 7593 prsradd 7594 prsrlt 7595 ltpsrprg 7611 map2psrprg 7613 pitonnlem1p1 7654 pitoregt0 7657 recidpirqlemcalc 7665

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