mulcomprg - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mulcomprg		GIF version

Theorem mulcomprg 7581

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 7476	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		elprnql 7482	. . . . . . . . 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 7476	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7482	. . . . . . . . . . . . 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 7384	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → (𝑧 ·_Q 𝑦) = (𝑦 ·_Q 𝑧))
8	7	eqeq2d 2189	. . . . . . . . . . . 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 2474	. . . . . . . . 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 2474	. . . . 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 2641	. . . . 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 2728	. . 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 7483	. . . . . . . . 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 7483	. . . . . . . . . . . . 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 2474	. . . . . . . . 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 2474	. . . . 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 2641	. . . . 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 2728	. . 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 3788	. 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 7540	. . 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 7540	. 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 2221	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 1353 ∈ wcel 2148 ∃wrex 2456 {crab 2459 ⟨cop 3597 ‘cfv 5218 (class class class)co 5877 1^st c1st 6141 2^nd c2nd 6142 Qcnq 7281 ·_Q cmq 7284 Pcnp 7292 ·_P cmp 7295

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-mi 7307 df-mpq 7346 df-enq 7348 df-nqqs 7349 df-mqqs 7351 df-inp 7467 df-imp 7470

This theorem is referenced by: ltmprr 7643 mulcmpblnrlemg 7741 mulcomsrg 7758 mulasssrg 7759 m1m1sr 7762 recexgt0sr 7774 mulgt0sr 7779 mulextsr1lem 7781 recidpirqlemcalc 7858

W3C validator