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Theorem addcomprg 7691

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 7588	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7594	. . . . . . . . 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 7588	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7594	. . . . . . . . . . . . 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 7494	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +_Q 𝑧) = (𝑧 +_Q 𝑦))
8	7	eqeq2d 2217	. . . . . . . . . . . 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 2503	. . . . . . . . 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 2503	. . . . 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 2670	. . . . 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 2761	. . 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 7595	. . . . . . . . 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 7595	. . . . . . . . . . . . 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 2503	. . . . . . . . 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 2503	. . . . 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 2670	. . . . 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 2761	. . 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 3827	. 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 7651	. . 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 7651	. 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 2249	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 1373 ∈ wcel 2176 ∃wrex 2485 {crab 2488 ⟨cop 3636 ‘cfv 5271 (class class class)co 5944 1^st c1st 6224 2^nd c2nd 6225 Qcnq 7393 +_Q cplq 7395 Pcnp 7404 +_P cpp 7406

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-iord 4413 df-on 4415 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-oadd 6506 df-omul 6507 df-er 6620 df-ec 6622 df-qs 6626 df-ni 7417 df-pli 7418 df-mi 7419 df-plpq 7457 df-enq 7460 df-nqqs 7461 df-plqqs 7462 df-inp 7579 df-iplp 7581

This theorem is referenced by: prplnqu 7733 addextpr 7734 caucvgprlemcanl 7757 caucvgprprlemnkltj 7802 caucvgprprlemnbj 7806 caucvgprprlemmu 7808 caucvgprprlemloc 7816 caucvgprprlemexbt 7819 caucvgprprlemexb 7820 caucvgprprlemaddq 7821 enrer 7848 addcmpblnr 7852 mulcmpblnrlemg 7853 ltsrprg 7860 addcomsrg 7868 mulcomsrg 7870 mulasssrg 7871 distrsrg 7872 lttrsr 7875 ltposr 7876 ltsosr 7877 0lt1sr 7878 0idsr 7880 1idsr 7881 ltasrg 7883 recexgt0sr 7886 mulgt0sr 7891 aptisr 7892 mulextsr1lem 7893 archsr 7895 srpospr 7896 prsrpos 7898 prsradd 7899 prsrlt 7900 ltpsrprg 7916 map2psrprg 7918 pitonnlem1p1 7959 pitoregt0 7962 recidpirqlemcalc 7970

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