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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 11177. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axi2m1 | ⊢ ((i · i) + 1) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 11074 | . . . . . 6 ⊢ 0R ∈ R | |
2 | 1sr 11075 | . . . . . 6 ⊢ 1R ∈ R | |
3 | mulcnsr 11130 | . . . . . 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 690 | . . . . 5 ⊢ (⟨0R, 1R⟩ · ⟨0R, 1R⟩) = ⟨((0R ·R 0R) +R (-1R ·R (1R ·R 1R))), ((1R ·R 0R) +R (0R ·R 1R))⟩ |
5 | 00sr 11093 | . . . . . . . . 9 ⊢ (0R ∈ R → (0R ·R 0R) = 0R) | |
6 | 1, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (0R ·R 0R) = 0R |
7 | 1idsr 11092 | . . . . . . . . . . 11 ⊢ (1R ∈ R → (1R ·R 1R) = 1R) | |
8 | 2, 7 | ax-mp 5 | . . . . . . . . . 10 ⊢ (1R ·R 1R) = 1R |
9 | 8 | oveq2i 7415 | . . . . . . . . 9 ⊢ (-1R ·R (1R ·R 1R)) = (-1R ·R 1R) |
10 | m1r 11076 | . . . . . . . . . 10 ⊢ -1R ∈ R | |
11 | 1idsr 11092 | . . . . . . . . . 10 ⊢ (-1R ∈ R → (-1R ·R 1R) = -1R) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . . 9 ⊢ (-1R ·R 1R) = -1R |
13 | 9, 12 | eqtri 2754 | . . . . . . . 8 ⊢ (-1R ·R (1R ·R 1R)) = -1R |
14 | 6, 13 | oveq12i 7416 | . . . . . . 7 ⊢ ((0R ·R 0R) +R (-1R ·R (1R ·R 1R))) = (0R +R -1R) |
15 | addcomsr 11081 | . . . . . . 7 ⊢ (0R +R -1R) = (-1R +R 0R) | |
16 | 0idsr 11091 | . . . . . . . 8 ⊢ (-1R ∈ R → (-1R +R 0R) = -1R) | |
17 | 10, 16 | ax-mp 5 | . . . . . . 7 ⊢ (-1R +R 0R) = -1R |
18 | 14, 15, 17 | 3eqtri 2758 | . . . . . 6 ⊢ ((0R ·R 0R) +R (-1R ·R (1R ·R 1R))) = -1R |
19 | 00sr 11093 | . . . . . . . . 9 ⊢ (1R ∈ R → (1R ·R 0R) = 0R) | |
20 | 2, 19 | ax-mp 5 | . . . . . . . 8 ⊢ (1R ·R 0R) = 0R |
21 | 1idsr 11092 | . . . . . . . . 9 ⊢ (0R ∈ R → (0R ·R 1R) = 0R) | |
22 | 1, 21 | ax-mp 5 | . . . . . . . 8 ⊢ (0R ·R 1R) = 0R |
23 | 20, 22 | oveq12i 7416 | . . . . . . 7 ⊢ ((1R ·R 0R) +R (0R ·R 1R)) = (0R +R 0R) |
24 | 0idsr 11091 | . . . . . . . 8 ⊢ (0R ∈ R → (0R +R 0R) = 0R) | |
25 | 1, 24 | ax-mp 5 | . . . . . . 7 ⊢ (0R +R 0R) = 0R |
26 | 23, 25 | eqtri 2754 | . . . . . 6 ⊢ ((1R ·R 0R) +R (0R ·R 1R)) = 0R |
27 | 18, 26 | opeq12i 4873 | . . . . 5 ⊢ ⟨((0R ·R 0R) +R (-1R ·R (1R ·R 1R))), ((1R ·R 0R) +R (0R ·R 1R))⟩ = ⟨-1R, 0R⟩ |
28 | 4, 27 | eqtri 2754 | . . . 4 ⊢ (⟨0R, 1R⟩ · ⟨0R, 1R⟩) = ⟨-1R, 0R⟩ |
29 | 28 | oveq1i 7414 | . . 3 ⊢ ((⟨0R, 1R⟩ · ⟨0R, 1R⟩) + ⟨1R, 0R⟩) = (⟨-1R, 0R⟩ + ⟨1R, 0R⟩) |
30 | addresr 11132 | . . . 4 ⊢ ((-1R ∈ R ∧ 1R ∈ R) → (⟨-1R, 0R⟩ + ⟨1R, 0R⟩) = ⟨(-1R +R 1R), 0R⟩) | |
31 | 10, 2, 30 | mp2an 689 | . . 3 ⊢ (⟨-1R, 0R⟩ + ⟨1R, 0R⟩) = ⟨(-1R +R 1R), 0R⟩ |
32 | m1p1sr 11086 | . . . 4 ⊢ (-1R +R 1R) = 0R | |
33 | 32 | opeq1i 4871 | . . 3 ⊢ ⟨(-1R +R 1R), 0R⟩ = ⟨0R, 0R⟩ |
34 | 29, 31, 33 | 3eqtri 2758 | . 2 ⊢ ((⟨0R, 1R⟩ · ⟨0R, 1R⟩) + ⟨1R, 0R⟩) = ⟨0R, 0R⟩ |
35 | df-i 11118 | . . . 4 ⊢ i = ⟨0R, 1R⟩ | |
36 | 35, 35 | oveq12i 7416 | . . 3 ⊢ (i · i) = (⟨0R, 1R⟩ · ⟨0R, 1R⟩) |
37 | df-1 11117 | . . 3 ⊢ 1 = ⟨1R, 0R⟩ | |
38 | 36, 37 | oveq12i 7416 | . 2 ⊢ ((i · i) + 1) = ((⟨0R, 1R⟩ · ⟨0R, 1R⟩) + ⟨1R, 0R⟩) |
39 | df-0 11116 | . 2 ⊢ 0 = ⟨0R, 0R⟩ | |
40 | 34, 38, 39 | 3eqtr4i 2764 | 1 ⊢ ((i · i) + 1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ⟨cop 4629 (class class class)co 7404 Rcnr 10859 0Rc0r 10860 1Rc1r 10861 -1Rcm1r 10862 +R cplr 10863 ·R cmr 10864 0cc0 11109 1c1 11110 ici 11111 + caddc 11112 · cmul 11114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-omul 8469 df-er 8702 df-ec 8704 df-qs 8708 df-ni 10866 df-pli 10867 df-mi 10868 df-lti 10869 df-plpq 10902 df-mpq 10903 df-ltpq 10904 df-enq 10905 df-nq 10906 df-erq 10907 df-plq 10908 df-mq 10909 df-1nq 10910 df-rq 10911 df-ltnq 10912 df-np 10975 df-1p 10976 df-plp 10977 df-mp 10978 df-ltp 10979 df-enr 11049 df-nr 11050 df-plr 11051 df-mr 11052 df-0r 11054 df-1r 11055 df-m1r 11056 df-c 11115 df-0 11116 df-1 11117 df-i 11118 df-add 11120 df-mul 11121 |
This theorem is referenced by: (None) |
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