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Theorem mulcomprg 7642

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 7537	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7543	. . . . . . . . 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 7537	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7543	. . . . . . . . . . . . 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 7445	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → (𝑧 ·_Q 𝑦) = (𝑦 ·_Q 𝑧))
8	7	eqeq2d 2205	. . . . . . . . . . . 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 2491	. . . . . . . . 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 2491	. . . . 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 2658	. . . . 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 2749	. . 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 7544	. . . . . . . . 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 7544	. . . . . . . . . . . . 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 2491	. . . . . . . . 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 2491	. . . . 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 2658	. . . . 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 2749	. . 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 3813	. 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 7601	. . 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 7601	. 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 2237	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 1364 ∈ wcel 2164 ∃wrex 2473 {crab 2476 ⟨cop 3622 ‘cfv 5255 (class class class)co 5919 1^st c1st 6193 2^nd c2nd 6194 Qcnq 7342 ·_Q cmq 7345 Pcnp 7353 ·_P cmp 7356

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-iord 4398 df-on 4400 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-oadd 6475 df-omul 6476 df-er 6589 df-ec 6591 df-qs 6595 df-ni 7366 df-mi 7368 df-mpq 7407 df-enq 7409 df-nqqs 7410 df-mqqs 7412 df-inp 7528 df-imp 7531

This theorem is referenced by: ltmprr 7704 mulcmpblnrlemg 7802 mulcomsrg 7819 mulasssrg 7820 m1m1sr 7823 recexgt0sr 7835 mulgt0sr 7840 mulextsr1lem 7842 recidpirqlemcalc 7919

W3C validator