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Mirrors > Home > ILE Home > Th. List > crap0 | GIF version |
Description: The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.) |
Ref | Expression |
---|---|
crap0 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 # 0 ∨ 𝐵 # 0) ↔ (𝐴 + (i · 𝐵)) # 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7956 | . . 3 ⊢ 0 ∈ ℝ | |
2 | apreim 8558 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ∈ ℝ ∧ 0 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) # (0 + (i · 0)) ↔ (𝐴 # 0 ∨ 𝐵 # 0))) | |
3 | 1, 1, 2 | mpanr12 439 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + (i · 𝐵)) # (0 + (i · 0)) ↔ (𝐴 # 0 ∨ 𝐵 # 0))) |
4 | ax-icn 7905 | . . . . . 6 ⊢ i ∈ ℂ | |
5 | 4 | mul01i 8346 | . . . . 5 ⊢ (i · 0) = 0 |
6 | 5 | oveq2i 5885 | . . . 4 ⊢ (0 + (i · 0)) = (0 + 0) |
7 | 00id 8096 | . . . 4 ⊢ (0 + 0) = 0 | |
8 | 6, 7 | eqtri 2198 | . . 3 ⊢ (0 + (i · 0)) = 0 |
9 | 8 | breq2i 4011 | . 2 ⊢ ((𝐴 + (i · 𝐵)) # (0 + (i · 0)) ↔ (𝐴 + (i · 𝐵)) # 0) |
10 | 3, 9 | bitr3di 195 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 # 0 ∨ 𝐵 # 0) ↔ (𝐴 + (i · 𝐵)) # 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℝcr 7809 0cc0 7810 ici 7812 + caddc 7813 · cmul 7815 # cap 8536 |
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-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 |
This theorem depends on definitions: df-bi 117 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-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 |
This theorem is referenced by: abs00ap 11066 |
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