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Theorem mulcomprg 7592

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 7487	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7493	. . . . . . . . 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 7487	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7493	. . . . . . . . . . . . 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 7395	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → (𝑧 ·_Q 𝑦) = (𝑦 ·_Q 𝑧))
8	7	eqeq2d 2199	. . . . . . . . . . . 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 2484	. . . . . . . . 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 2484	. . . . 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 2651	. . . . 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 2738	. . 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 7494	. . . . . . . . 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 7494	. . . . . . . . . . . . 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 2484	. . . . . . . . 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 2484	. . . . 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 2651	. . . . 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 2738	. . 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 3798	. 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 7551	. . 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 7551	. 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 2231	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 1363 ∈ wcel 2158 ∃wrex 2466 {crab 2469 ⟨cop 3607 ‘cfv 5228 (class class class)co 5888 1^st c1st 6152 2^nd c2nd 6153 Qcnq 7292 ·_Q cmq 7295 Pcnp 7303 ·_P cmp 7306

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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-oadd 6434 df-omul 6435 df-er 6548 df-ec 6550 df-qs 6554 df-ni 7316 df-mi 7318 df-mpq 7357 df-enq 7359 df-nqqs 7360 df-mqqs 7362 df-inp 7478 df-imp 7481

This theorem is referenced by: ltmprr 7654 mulcmpblnrlemg 7752 mulcomsrg 7769 mulasssrg 7770 m1m1sr 7773 recexgt0sr 7785 mulgt0sr 7790 mulextsr1lem 7792 recidpirqlemcalc 7869

W3C validator