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Theorem mulcomprg 7778

Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)

**Assertion**
Ref	Expression
mulcomprg	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) = (𝐵 ·_P 𝐴))

Proof of Theorem mulcomprg Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7673	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7679	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐵)) → 𝑧 ∈ Q)
3	1, 2	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐵)) → 𝑧 ∈ Q)
4		prop 7673	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7679	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
6	4, 5	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
7		mulcomnqg 7581	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → (𝑧 ·_Q 𝑦) = (𝑦 ·_Q 𝑧))
8	7	eqeq2d 2241	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
9	6, 8	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴))) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
10	9	anassrs 400	. . . . . . . . . 10 ⊢ (((𝑧 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑦 ∈ (1^st ‘𝐴)) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
11	10	rexbidva 2527	. . . . . . . . 9 ⊢ ((𝑧 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
12	11	ancoms 268	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ Q) → (∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
13	3, 12	sylan2 286	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐵))) → (∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
14	13	anassrs 400	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑧 ∈ (1^st ‘𝐵)) → (∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
15	14	rexbidva 2527	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
16		rexcom 2695	. . . . 5 ⊢ (∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧) ↔ ∃𝑦 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧))
17	15, 16	bitrdi 196	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)))
18	17	rabbidv 2788	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)})
19		elprnqu 7680	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐵)) → 𝑧 ∈ Q)
20	1, 19	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐵)) → 𝑧 ∈ Q)
21		elprnqu 7680	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
22	4, 21	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
23	22, 8	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴))) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
24	23	anassrs 400	. . . . . . . . . 10 ⊢ (((𝑧 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
25	24	rexbidva 2527	. . . . . . . . 9 ⊢ ((𝑧 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
26	25	ancoms 268	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ Q) → (∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
27	20, 26	sylan2 286	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐵))) → (∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
28	27	anassrs 400	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑧 ∈ (2^nd ‘𝐵)) → (∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
29	28	rexbidva 2527	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
30		rexcom 2695	. . . . 5 ⊢ (∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧) ↔ ∃𝑦 ∈ (2^nd ‘𝐴)∃𝑧 ∈ (2^nd ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧))
31	29, 30	bitrdi 196	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (2^nd ‘𝐴)∃𝑧 ∈ (2^nd ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)))
32	31	rabbidv 2788	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → {𝑥 ∈ Q ∣ ∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)} = {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2^nd ‘𝐴)∃𝑧 ∈ (2^nd ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)})
33	18, 32	opeq12d 3865	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ⟨{𝑥 ∈ Q ∣ ∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)}, {𝑥 ∈ Q ∣ ∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)}⟩ = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2^nd ‘𝐴)∃𝑧 ∈ (2^nd ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)}⟩)
34		mpvlu 7737	. . 3 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·_P 𝐴) = ⟨{𝑥 ∈ Q ∣ ∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)}, {𝑥 ∈ Q ∣ ∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)}⟩)
35	34	ancoms 268	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐵 ·_P 𝐴) = ⟨{𝑥 ∈ Q ∣ ∃𝑧 ∈ (1^st ‘𝐵)∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)}, {𝑥 ∈ Q ∣ ∃𝑧 ∈ (2^nd ‘𝐵)∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦)}⟩)
36		mpvlu 7737	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2^nd ‘𝐴)∃𝑧 ∈ (2^nd ‘𝐵)𝑥 = (𝑦 ·_Q 𝑧)}⟩)
37	33, 35, 36	3eqtr4rd 2273	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 1395 ∈ wcel 2200 ∃wrex 2509 {crab 2512 ⟨cop 3669 ‘cfv 5318 (class class class)co 6007 1^st c1st 6290 2^nd c2nd 6291 Qcnq 7478 ·_Q cmq 7481 Pcnp 7489 ·_P cmp 7492

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-oadd 6572 df-omul 6573 df-er 6688 df-ec 6690 df-qs 6694 df-ni 7502 df-mi 7504 df-mpq 7543 df-enq 7545 df-nqqs 7546 df-mqqs 7548 df-inp 7664 df-imp 7667

This theorem is referenced by: ltmprr 7840 mulcmpblnrlemg 7938 mulcomsrg 7955 mulasssrg 7956 m1m1sr 7959 recexgt0sr 7971 mulgt0sr 7976 mulextsr1lem 7978 recidpirqlemcalc 8055

W3C validator