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Mirrors > Home > ILE Home > Th. List > distrprg | GIF version |
Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Ref | Expression |
---|---|
distrprg | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distrlem1prl 7062 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))) | |
2 | distrlem5prl 7066 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))) | |
3 | 1, 2 | eqssd 3029 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) = (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))) |
4 | distrlem1pru 7063 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))) | |
5 | distrlem5pru 7067 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) | |
6 | 4, 5 | eqssd 3029 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) = (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))) |
7 | simp1 941 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ∈ P) | |
8 | simp2 942 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐵 ∈ P) | |
9 | simp3 943 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐶 ∈ P) | |
10 | addclpr 7017 | . . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P) | |
11 | 8, 9, 10 | syl2anc 403 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P) |
12 | mulclpr 7052 | . . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P) | |
13 | 7, 11, 12 | syl2anc 403 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P) |
14 | mulclpr 7052 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P) | |
15 | 7, 8, 14 | syl2anc 403 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P) |
16 | mulclpr 7052 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P) | |
17 | 7, 9, 16 | syl2anc 403 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P) |
18 | addclpr 7017 | . . . 4 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ∈ P) | |
19 | 15, 17, 18 | syl2anc 403 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ∈ P) |
20 | preqlu 6952 | . . 3 ⊢ (((𝐴 ·P (𝐵 +P 𝐶)) ∈ P ∧ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ∈ P) → ((𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ↔ ((1st ‘(𝐴 ·P (𝐵 +P 𝐶))) = (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ∧ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) = (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))) | |
21 | 13, 19, 20 | syl2anc 403 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ↔ ((1st ‘(𝐴 ·P (𝐵 +P 𝐶))) = (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ∧ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) = (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))) |
22 | 3, 6, 21 | mpbir2and 888 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 922 = wceq 1287 ∈ wcel 1436 ‘cfv 4972 (class class class)co 5594 1st c1st 5847 2nd c2nd 5848 Pcnp 6771 +P cpp 6773 ·P cmp 6774 |
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 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 |
This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-eprel 4083 df-id 4087 df-po 4090 df-iso 4091 df-iord 4160 df-on 4162 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-recs 6005 df-irdg 6070 df-1o 6116 df-2o 6117 df-oadd 6120 df-omul 6121 df-er 6225 df-ec 6227 df-qs 6231 df-ni 6784 df-pli 6785 df-mi 6786 df-lti 6787 df-plpq 6824 df-mpq 6825 df-enq 6827 df-nqqs 6828 df-plqqs 6829 df-mqqs 6830 df-1nqqs 6831 df-rq 6832 df-ltnqqs 6833 df-enq0 6904 df-nq0 6905 df-0nq0 6906 df-plq0 6907 df-mq0 6908 df-inp 6946 df-iplp 6948 df-imp 6949 |
This theorem is referenced by: ltmprr 7122 mulcmpblnrlemg 7207 mulasssrg 7225 distrsrg 7226 m1m1sr 7228 1idsr 7235 recexgt0sr 7240 mulgt0sr 7244 mulextsr1lem 7246 recidpirqlemcalc 7315 |
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