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Theorem addcomprg 7410

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 7307	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7313	. . . . . . . . 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 7307	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7313	. . . . . . . . . . . . 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 7213	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) = (𝑧 +_Q 𝑦))
8	7	eqeq2d 2152	. . . . . . . . . . . 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 398	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑧 ∈ (1^st ‘𝐴)) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
11	10	rexbidva 2435	. . . . . . . . 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 398	. . . . . 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 2435	. . . . 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 2598	. . . . 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 2678	. . 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 7314	. . . . . . . . 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 7314	. . . . . . . . . . . . 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 398	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑧 ∈ (2^nd ‘𝐴)) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
25	24	rexbidva 2435	. . . . . . . . 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 398	. . . . . 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 2435	. . . . 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 2598	. . . . 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 2678	. . 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 3721	. 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 7370	. . 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 7370	. 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 2184	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 1332 ∈ wcel 1481 ∃wrex 2418 {crab 2421 ⟨cop 3535 ‘cfv 5131 (class class class)co 5782 1^st c1st 6044 2^nd c2nd 6045 Qcnq 7112 +_Q cplq 7114 Pcnp 7123 +_P cpp 7125

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-plpq 7176 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-inp 7298 df-iplp 7300

This theorem is referenced by: prplnqu 7452 addextpr 7453 caucvgprlemcanl 7476 caucvgprprlemnkltj 7521 caucvgprprlemnbj 7525 caucvgprprlemmu 7527 caucvgprprlemloc 7535 caucvgprprlemexbt 7538 caucvgprprlemexb 7539 caucvgprprlemaddq 7540 enrer 7567 addcmpblnr 7571 mulcmpblnrlemg 7572 ltsrprg 7579 addcomsrg 7587 mulcomsrg 7589 mulasssrg 7590 distrsrg 7591 lttrsr 7594 ltposr 7595 ltsosr 7596 0lt1sr 7597 0idsr 7599 1idsr 7600 ltasrg 7602 recexgt0sr 7605 mulgt0sr 7610 aptisr 7611 mulextsr1lem 7612 archsr 7614 srpospr 7615 prsrpos 7617 prsradd 7618 prsrlt 7619 ltpsrprg 7635 map2psrprg 7637 pitonnlem1p1 7678 pitoregt0 7681 recidpirqlemcalc 7689

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