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Mirrors > Home > MPE Home > Th. List > addcomnq | Structured version Visualization version GIF version |
Description: Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addcomnq | ⊢ (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcompq 9985 | . . . 4 ⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴) | |
2 | 1 | fveq2i 6357 | . . 3 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘(𝐵 +pQ 𝐴)) |
3 | addpqnq 9973 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) | |
4 | addpqnq 9973 | . . . 4 ⊢ ((𝐵 ∈ Q ∧ 𝐴 ∈ Q) → (𝐵 +Q 𝐴) = ([Q]‘(𝐵 +pQ 𝐴))) | |
5 | 4 | ancoms 468 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐴) = ([Q]‘(𝐵 +pQ 𝐴))) |
6 | 2, 3, 5 | 3eqtr4a 2821 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)) |
7 | addnqf 9983 | . . . 4 ⊢ +Q :(Q × Q)⟶Q | |
8 | 7 | fdmi 6214 | . . 3 ⊢ dom +Q = (Q × Q) |
9 | 8 | ndmovcom 6988 | . 2 ⊢ (¬ (𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)) |
10 | 6, 9 | pm2.61i 176 | 1 ⊢ (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1632 ∈ wcel 2140 × cxp 5265 ‘cfv 6050 (class class class)co 6815 +pQ cplpq 9883 Qcnq 9887 [Q]cerq 9889 +Q cplq 9890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-omul 7736 df-er 7914 df-ni 9907 df-pli 9908 df-mi 9909 df-lti 9910 df-plpq 9943 df-enq 9946 df-nq 9947 df-erq 9948 df-plq 9949 df-1nq 9951 |
This theorem is referenced by: ltaddnq 10009 addclprlem2 10052 addclpr 10053 addcompr 10056 distrlem4pr 10061 prlem934 10068 ltexprlem2 10072 ltexprlem6 10076 ltexprlem7 10077 prlem936 10082 |
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