![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > negexsr | Structured version Visualization version GIF version |
Description: Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
negexsr | ⊢ (𝐴 ∈ R → ∃𝑥 ∈ R (𝐴 +R 𝑥) = 0R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m1r 10220 | . . 3 ⊢ -1R ∈ R | |
2 | mulclsr 10222 | . . 3 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
3 | 1, 2 | mpan2 684 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R -1R) ∈ R) |
4 | pn0sr 10239 | . 2 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) | |
5 | oveq2 6914 | . . . 4 ⊢ (𝑥 = (𝐴 ·R -1R) → (𝐴 +R 𝑥) = (𝐴 +R (𝐴 ·R -1R))) | |
6 | 5 | eqeq1d 2828 | . . 3 ⊢ (𝑥 = (𝐴 ·R -1R) → ((𝐴 +R 𝑥) = 0R ↔ (𝐴 +R (𝐴 ·R -1R)) = 0R)) |
7 | 6 | rspcev 3527 | . 2 ⊢ (((𝐴 ·R -1R) ∈ R ∧ (𝐴 +R (𝐴 ·R -1R)) = 0R) → ∃𝑥 ∈ R (𝐴 +R 𝑥) = 0R) |
8 | 3, 4, 7 | syl2anc 581 | 1 ⊢ (𝐴 ∈ R → ∃𝑥 ∈ R (𝐴 +R 𝑥) = 0R) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∃wrex 3119 (class class class)co 6906 Rcnr 10003 0Rc0r 10004 -1Rcm1r 10006 +R cplr 10007 ·R cmr 10008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-omul 7832 df-er 8010 df-ec 8012 df-qs 8016 df-ni 10010 df-pli 10011 df-mi 10012 df-lti 10013 df-plpq 10046 df-mpq 10047 df-ltpq 10048 df-enq 10049 df-nq 10050 df-erq 10051 df-plq 10052 df-mq 10053 df-1nq 10054 df-rq 10055 df-ltnq 10056 df-np 10119 df-1p 10120 df-plp 10121 df-mp 10122 df-ltp 10123 df-enr 10193 df-nr 10194 df-plr 10195 df-mr 10196 df-0r 10198 df-1r 10199 df-m1r 10200 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |