![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > re1m1e0m0 | Structured version Visualization version GIF version |
Description: Equality of two left-additive identities. See resubidaddlid 42402. Uses ax-i2m1 11221. (Contributed by SN, 25-Dec-2023.) |
Ref | Expression |
---|---|
re1m1e0m0 | ⊢ (1 −ℝ 1) = (0 −ℝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 11262 | . . 3 ⊢ (⊤ → 0 ∈ ℝ) | |
2 | 1re 11259 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rersubcl 42385 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (1 −ℝ 1) ∈ ℝ) | |
4 | 2, 2, 3 | mp2an 692 | . . . 4 ⊢ (1 −ℝ 1) ∈ ℝ |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (1 −ℝ 1) ∈ ℝ) |
6 | ax-icn 11212 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | 6, 6 | mulcli 11266 | . . . . . . 7 ⊢ (i · i) ∈ ℂ |
8 | ax-1cn 11211 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | 4 | recni 11273 | . . . . . . 7 ⊢ (1 −ℝ 1) ∈ ℂ |
10 | 7, 8, 9 | addassi 11269 | . . . . . 6 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + (1 + (1 −ℝ 1))) |
11 | repncan3 42390 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (1 + (1 −ℝ 1)) = 1) | |
12 | 2, 2, 11 | mp2an 692 | . . . . . . 7 ⊢ (1 + (1 −ℝ 1)) = 1 |
13 | 12 | oveq2i 7442 | . . . . . 6 ⊢ ((i · i) + (1 + (1 −ℝ 1))) = ((i · i) + 1) |
14 | 10, 13 | eqtri 2763 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + 1) |
15 | ax-i2m1 11221 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
16 | 15 | oveq1i 7441 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = (0 + (1 −ℝ 1)) |
17 | 14, 16, 15 | 3eqtr3i 2771 | . . . 4 ⊢ (0 + (1 −ℝ 1)) = 0 |
18 | 17 | a1i 11 | . . 3 ⊢ (⊤ → (0 + (1 −ℝ 1)) = 0) |
19 | 1, 5, 18 | reladdrsub 42392 | . 2 ⊢ (⊤ → (1 −ℝ 1) = (0 −ℝ 0)) |
20 | 19 | mptru 1544 | 1 ⊢ (1 −ℝ 1) = (0 −ℝ 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 ici 11155 + caddc 11156 · cmul 11158 −ℝ cresub 42372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-addass 11218 ax-i2m1 11221 ax-1ne0 11222 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-resub 42373 |
This theorem is referenced by: sn-00idlem1 42405 sn-00idlem2 42406 remul02 42412 |
Copyright terms: Public domain | W3C validator |