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Mirrors > Home > MPE Home > Th. List > mulcompr | Structured version Visualization version GIF version |
Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulcompr | ⊢ (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpv 9871 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)}) | |
2 | mpv 9871 | . . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧)}) | |
3 | mulcomnq 9813 | . . . . . . . . 9 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦) | |
4 | 3 | eqeq2i 2663 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 ·Q 𝑧) ↔ 𝑥 = (𝑧 ·Q 𝑦)) |
5 | 4 | 2rexbii 3071 | . . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 ·Q 𝑦)) |
6 | rexcom 3128 | . . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)) | |
7 | 5, 6 | bitri 264 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)) |
8 | 7 | abbii 2768 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧)} = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)} |
9 | 2, 8 | syl6eq 2701 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)}) |
10 | 9 | ancoms 468 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)}) |
11 | 1, 10 | eqtr4d 2688 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)) |
12 | dmmp 9873 | . . 3 ⊢ dom ·P = (P × P) | |
13 | 12 | ndmovcom 6863 | . 2 ⊢ (¬ (𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)) |
14 | 11, 13 | pm2.61i 176 | 1 ⊢ (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1523 ∈ wcel 2030 {cab 2637 ∃wrex 2942 (class class class)co 6690 ·Q cmq 9716 Pcnp 9719 ·P cmp 9722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-omul 7610 df-er 7787 df-ni 9732 df-mi 9734 df-lti 9735 df-mpq 9769 df-enq 9771 df-nq 9772 df-erq 9773 df-mq 9775 df-1nq 9776 df-np 9841 df-mp 9844 |
This theorem is referenced by: mulcmpblnrlem 9929 mulcomsr 9948 mulasssr 9949 m1m1sr 9952 recexsrlem 9962 mulgt0sr 9964 |
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