Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0cnALT2 | Structured version Visualization version GIF version |
Description: Alternate proof of 0cnALT 10867 which is shorter, but depends on ax-8 2115, ax-13 2389, ax-sep 5196, ax-nul 5203, ax-pow 5259, ax-pr 5323, ax-un 7454, and every complex number axiom except ax-pre-mulgt0 10607 and ax-pre-sup 10608. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0cnALT2 | ⊢ 0 ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 10589 | . . 3 ⊢ i ∈ ℂ | |
2 | cnegex 10814 | . . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℂ (i + 𝑥) = 0 |
4 | addcl 10612 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ) | |
5 | 1, 4 | mpan 688 | . . . 4 ⊢ (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ) |
6 | eleq1 2899 | . . . 4 ⊢ ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ)) | |
7 | 5, 6 | syl5ibcom 247 | . . 3 ⊢ (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ)) |
8 | 7 | rexlimiv 3279 | . 2 ⊢ (∃𝑥 ∈ ℂ (i + 𝑥) = 0 → 0 ∈ ℂ) |
9 | 3, 8 | ax-mp 5 | 1 ⊢ 0 ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ∃wrex 3138 (class class class)co 7149 ℂcc 10528 0cc0 10530 ici 10532 + caddc 10533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-ltxr 10673 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |