![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axi2m1 | Structured version Visualization version GIF version |
Description: i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 11126. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axi2m1 | ⊢ ((i · i) + 1) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 11023 | . . . . . 6 ⊢ 0R ∈ R | |
2 | 1sr 11024 | . . . . . 6 ⊢ 1R ∈ R | |
3 | mulcnsr 11079 | . . . . . 6 ⊢ (((0R ∈ R ∧ 1R ∈ R) ∧ (0R ∈ R ∧ 1R ∈ R)) → (⟨0R, 1R⟩ · ⟨0R, 1R⟩) = ⟨((0R ·R 0R) +R (-1R ·R (1R ·R 1R))), ((1R ·R 0R) +R (0R ·R 1R))⟩) | |
4 | 1, 2, 1, 2, 3 | mp4an 692 | . . . . 5 ⊢ (⟨0R, 1R⟩ · ⟨0R, 1R⟩) = ⟨((0R ·R 0R) +R (-1R ·R (1R ·R 1R))), ((1R ·R 0R) +R (0R ·R 1R))⟩ |
5 | 00sr 11042 | . . . . . . . . 9 ⊢ (0R ∈ R → (0R ·R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (0R ·R 0R) = 0R |
7 | 1idsr 11041 | . . . . . . . . . . 11 ⊢ (1R ∈ R → (1R ·R 1R) = 1R) | |
8 | 2, 7 | ax-mp 5 | . . . . . . . . . 10 ⊢ (1R ·R 1R) = 1R |
9 | 8 | oveq2i 7373 | . . . . . . . . 9 ⊢ (-1R ·R (1R ·R 1R)) = (-1R ·R 1R) |
10 | m1r 11025 | . . . . . . . . . 10 ⊢ -1R ∈ R | |
11 | 1idsr 11041 | . . . . . . . . . 10 ⊢ (-1R ∈ R → (-1R ·R 1R) = -1R) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . . 9 ⊢ (-1R ·R 1R) = -1R |
13 | 9, 12 | eqtri 2765 | . . . . . . . 8 ⊢ (-1R ·R (1R ·R 1R)) = -1R |
14 | 6, 13 | oveq12i 7374 | . . . . . . 7 ⊢ ((0R ·R 0R) +R (-1R ·R (1R ·R 1R))) = (0R +R -1R) |
15 | addcomsr 11030 | . . . . . . 7 ⊢ (0R +R -1R) = (-1R +R 0R) | |
16 | 0idsr 11040 | . . . . . . . 8 ⊢ (-1R ∈ R → (-1R +R 0R) = -1R) | |
17 | 10, 16 | ax-mp 5 | . . . . . . 7 ⊢ (-1R +R 0R) = -1R |
18 | 14, 15, 17 | 3eqtri 2769 | . . . . . 6 ⊢ ((0R ·R 0R) +R (-1R ·R (1R ·R 1R))) = -1R |
19 | 00sr 11042 | . . . . . . . . 9 ⊢ (1R ∈ R → (1R ·R 0R) = 0R) | |
20 | 2, 19 | ax-mp 5 | . . . . . . . 8 ⊢ (1R ·R 0R) = 0R |
21 | 1idsr 11041 | . . . . . . . . 9 ⊢ (0R ∈ R → (0R ·R 1R) = 0R) | |
22 | 1, 21 | ax-mp 5 | . . . . . . . 8 ⊢ (0R ·R 1R) = 0R |
23 | 20, 22 | oveq12i 7374 | . . . . . . 7 ⊢ ((1R ·R 0R) +R (0R ·R 1R)) = (0R +R 0R) |
24 | 0idsr 11040 | . . . . . . . 8 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
25 | 1, 24 | ax-mp 5 | . . . . . . 7 ⊢ (0R +R 0R) = 0R |
26 | 23, 25 | eqtri 2765 | . . . . . 6 ⊢ ((1R ·R 0R) +R (0R ·R 1R)) = 0R |
27 | 18, 26 | opeq12i 4840 | . . . . 5 ⊢ ⟨((0R ·R 0R) +R (-1R ·R (1R ·R 1R))), ((1R ·R 0R) +R (0R ·R 1R))⟩ = ⟨-1R, 0R⟩ |
28 | 4, 27 | eqtri 2765 | . . . 4 ⊢ (⟨0R, 1R⟩ · ⟨0R, 1R⟩) = ⟨-1R, 0R⟩ |
29 | 28 | oveq1i 7372 | . . 3 ⊢ ((⟨0R, 1R⟩ · ⟨0R, 1R⟩) + ⟨1R, 0R⟩) = (⟨-1R, 0R⟩ + ⟨1R, 0R⟩) |
30 | addresr 11081 | . . . 4 ⊢ ((-1R ∈ R ∧ 1R ∈ R) → (⟨-1R, 0R⟩ + ⟨1R, 0R⟩) = ⟨(-1R +R 1R), 0R⟩) | |
31 | 10, 2, 30 | mp2an 691 | . . 3 ⊢ (⟨-1R, 0R⟩ + ⟨1R, 0R⟩) = ⟨(-1R +R 1R), 0R⟩ |
32 | m1p1sr 11035 | . . . 4 ⊢ (-1R +R 1R) = 0R | |
33 | 32 | opeq1i 4838 | . . 3 ⊢ ⟨(-1R +R 1R), 0R⟩ = ⟨0R, 0R⟩ |
34 | 29, 31, 33 | 3eqtri 2769 | . 2 ⊢ ((⟨0R, 1R⟩ · ⟨0R, 1R⟩) + ⟨1R, 0R⟩) = ⟨0R, 0R⟩ |
35 | df-i 11067 | . . . 4 ⊢ i = ⟨0R, 1R⟩ | |
36 | 35, 35 | oveq12i 7374 | . . 3 ⊢ (i · i) = (⟨0R, 1R⟩ · ⟨0R, 1R⟩) |
37 | df-1 11066 | . . 3 ⊢ 1 = ⟨1R, 0R⟩ | |
38 | 36, 37 | oveq12i 7374 | . 2 ⊢ ((i · i) + 1) = ((⟨0R, 1R⟩ · ⟨0R, 1R⟩) + ⟨1R, 0R⟩) |
39 | df-0 11065 | . 2 ⊢ 0 = ⟨0R, 0R⟩ | |
40 | 34, 38, 39 | 3eqtr4i 2775 | 1 ⊢ ((i · i) + 1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ⟨cop 4597 (class class class)co 7362 Rcnr 10808 0Rc0r 10809 1Rc1r 10810 -1Rcm1r 10811 +R cplr 10812 ·R cmr 10813 0cc0 11058 1c1 11059 ici 11060 + caddc 11061 · cmul 11063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ec 8657 df-qs 8661 df-ni 10815 df-pli 10816 df-mi 10817 df-lti 10818 df-plpq 10851 df-mpq 10852 df-ltpq 10853 df-enq 10854 df-nq 10855 df-erq 10856 df-plq 10857 df-mq 10858 df-1nq 10859 df-rq 10860 df-ltnq 10861 df-np 10924 df-1p 10925 df-plp 10926 df-mp 10927 df-ltp 10928 df-enr 10998 df-nr 10999 df-plr 11000 df-mr 11001 df-0r 11003 df-1r 11004 df-m1r 11005 df-c 11064 df-0 11065 df-1 11066 df-i 11067 df-add 11069 df-mul 11070 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |