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Theorem mulcomprg 6770

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 6665	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 6671	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐵)) → 𝑧 ∈ Q)
3	1, 2	sylan 277	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐵)) → 𝑧 ∈ Q)
4		prop 6665	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 6671	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
6	4, 5	sylan 277	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ Q)
7		mulcomnqg 6573	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → (𝑧 ·_Q 𝑦) = (𝑦 ·_Q 𝑧))
8	7	eqeq2d 2092	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
9	6, 8	sylan2 280	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴))) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
10	9	anassrs 392	. . . . . . . . . 10 ⊢ (((𝑧 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑦 ∈ (1^st ‘𝐴)) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
11	10	rexbidva 2365	. . . . . . . . 9 ⊢ ((𝑧 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (1^st ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
12	11	ancoms 264	. . . . . . . 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 280	. . . . . . 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 392	. . . . . 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 2365	. . . . 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 2518	. . . . 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	syl6bb 194	. . . 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 2593	. . 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 6672	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐵)) → 𝑧 ∈ Q)
20	1, 19	sylan 277	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑧 ∈ (2^nd ‘𝐵)) → 𝑧 ∈ Q)
21		elprnqu 6672	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
22	4, 21	sylan 277	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
23	22, 8	sylan2 280	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Q ∧ (𝐴 ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴))) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
24	23	anassrs 392	. . . . . . . . . 10 ⊢ (((𝑧 ∈ Q ∧ 𝐴 ∈ P) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → (𝑥 = (𝑧 ·_Q 𝑦) ↔ 𝑥 = (𝑦 ·_Q 𝑧)))
25	24	rexbidva 2365	. . . . . . . . 9 ⊢ ((𝑧 ∈ Q ∧ 𝐴 ∈ P) → (∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑧 ·_Q 𝑦) ↔ ∃𝑦 ∈ (2^nd ‘𝐴)𝑥 = (𝑦 ·_Q 𝑧)))
26	25	ancoms 264	. . . . . . . 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 280	. . . . . . 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 392	. . . . . 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 2365	. . . . 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 2518	. . . . 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	syl6bb 194	. . . 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 2593	. . 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 3578	. 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 6729	. . 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 264	. 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 6729	. 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 2124	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) = (𝐵 ·_P 𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 {crab 2352 ⟨cop 3401 ‘cfv 4922 (class class class)co 5532 1^st c1st 5785 2^nd c2nd 5786 Qcnq 6470 ·_Q cmq 6473 Pcnp 6481 ·_P cmp 6484

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-mqqs 6540 df-inp 6656 df-imp 6659

This theorem is referenced by: ltmprr 6832 mulcmpblnrlemg 6917 mulcomsrg 6934 mulasssrg 6935 m1m1sr 6938 recexgt0sr 6950 mulgt0sr 6954 mulextsr1lem 6956 recidpirqlemcalc 7025

W3C validator