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Theorem addcomprg 7841

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 7738	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7744	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
3	1, 2	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
4		prop 7738	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7744	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
6	4, 5	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
7		addcomnqg 7644	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) = (𝑧 +_Q 𝑦))
8	7	eqeq2d 2243	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
9	6, 8	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴))) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
10	9	anassrs 400	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑧 ∈ (1^st ‘𝐴)) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
11	10	rexbidva 2530	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (1^st ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
12	11	ancoms 268	. . . . . . . 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 286	. . . . . . 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 400	. . . . . 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 2530	. . . . 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 2698	. . . . 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 196	. . . 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 2792	. . 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 7745	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐵)) → 𝑦 ∈ Q)
20	1, 19	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐵)) → 𝑦 ∈ Q)
21		elprnqu 7745	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐴)) → 𝑧 ∈ Q)
22	4, 21	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐴)) → 𝑧 ∈ Q)
23	22, 8	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐴))) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
24	23	anassrs 400	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑧 ∈ (2^nd ‘𝐴)) → (𝑥 = (𝑦 +_Q 𝑧) ↔ 𝑥 = (𝑧 +_Q 𝑦)))
25	24	rexbidva 2530	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 +_Q 𝑧) ↔ ∃𝑧 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 +_Q 𝑦)))
26	25	ancoms 268	. . . . . . . 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 286	. . . . . . 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 400	. . . . . 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 2530	. . . . 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 2698	. . . . 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 196	. . . 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 2792	. . 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 3875	. 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 7801	. . 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 268	. 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 7801	. 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 2275	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) = (𝐵 +_P 𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2202 ∃wrex 2512 {crab 2515 ⟨cop 3676 ‘cfv 5333 (class class class)co 6028 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7543 +_Q cplq 7545 Pcnp 7554 +_P cpp 7556

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7567 df-pli 7568 df-mi 7569 df-plpq 7607 df-enq 7610 df-nqqs 7611 df-plqqs 7612 df-inp 7729 df-iplp 7731

This theorem is referenced by: prplnqu 7883 addextpr 7884 caucvgprlemcanl 7907 caucvgprprlemnkltj 7952 caucvgprprlemnbj 7956 caucvgprprlemmu 7958 caucvgprprlemloc 7966 caucvgprprlemexbt 7969 caucvgprprlemexb 7970 caucvgprprlemaddq 7971 enrer 7998 addcmpblnr 8002 mulcmpblnrlemg 8003 ltsrprg 8010 addcomsrg 8018 mulcomsrg 8020 mulasssrg 8021 distrsrg 8022 lttrsr 8025 ltposr 8026 ltsosr 8027 0lt1sr 8028 0idsr 8030 1idsr 8031 ltasrg 8033 recexgt0sr 8036 mulgt0sr 8041 aptisr 8042 mulextsr1lem 8043 archsr 8045 srpospr 8046 prsrpos 8048 prsradd 8049 prsrlt 8050 ltpsrprg 8066 map2psrprg 8068 pitonnlem1p1 8109 pitoregt0 8112 recidpirqlemcalc 8120

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