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Theorem addcomprg 7519

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 7416	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7422	. . . . . . . . 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 7416	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7422	. . . . . . . . . . . . 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 7322	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) = (𝑧 +_Q 𝑦))
8	7	eqeq2d 2177	. . . . . . . . . . . 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 2463	. . . . . . . . 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 2463	. . . . 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 2630	. . . . 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	bitrdi 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 2715	. . 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 7423	. . . . . . . . 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 7423	. . . . . . . . . . . . 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 2463	. . . . . . . . 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 2463	. . . . 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 2630	. . . . 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	bitrdi 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 2715	. . 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 3766	. 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 7479	. . 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 7479	. 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 2209	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 1343 ∈ wcel 2136 ∃wrex 2445 {crab 2448 ⟨cop 3579 ‘cfv 5188 (class class class)co 5842 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 +_Q cplq 7223 Pcnp 7232 +_P cpp 7234

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-oadd 6388 df-omul 6389 df-er 6501 df-ec 6503 df-qs 6507 df-ni 7245 df-pli 7246 df-mi 7247 df-plpq 7285 df-enq 7288 df-nqqs 7289 df-plqqs 7290 df-inp 7407 df-iplp 7409

This theorem is referenced by: prplnqu 7561 addextpr 7562 caucvgprlemcanl 7585 caucvgprprlemnkltj 7630 caucvgprprlemnbj 7634 caucvgprprlemmu 7636 caucvgprprlemloc 7644 caucvgprprlemexbt 7647 caucvgprprlemexb 7648 caucvgprprlemaddq 7649 enrer 7676 addcmpblnr 7680 mulcmpblnrlemg 7681 ltsrprg 7688 addcomsrg 7696 mulcomsrg 7698 mulasssrg 7699 distrsrg 7700 lttrsr 7703 ltposr 7704 ltsosr 7705 0lt1sr 7706 0idsr 7708 1idsr 7709 ltasrg 7711 recexgt0sr 7714 mulgt0sr 7719 aptisr 7720 mulextsr1lem 7721 archsr 7723 srpospr 7724 prsrpos 7726 prsradd 7727 prsrlt 7728 ltpsrprg 7744 map2psrprg 7746 pitonnlem1p1 7787 pitoregt0 7790 recidpirqlemcalc 7798

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