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Mirrors > Home > MPE Home > Th. List > Mathboxes > re1m1e0m0 | Structured version Visualization version GIF version |
Description: Equality of two left-additive identities. See resubidaddid1 40802. Uses ax-i2m1 11115. (Contributed by SN, 25-Dec-2023.) |
Ref | Expression |
---|---|
re1m1e0m0 | ⊢ (1 −ℝ 1) = (0 −ℝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 11154 | . . 3 ⊢ (⊤ → 0 ∈ ℝ) | |
2 | 1re 11151 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rersubcl 40785 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (1 −ℝ 1) ∈ ℝ) | |
4 | 2, 2, 3 | mp2an 690 | . . . 4 ⊢ (1 −ℝ 1) ∈ ℝ |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (1 −ℝ 1) ∈ ℝ) |
6 | ax-icn 11106 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | 6, 6 | mulcli 11158 | . . . . . . 7 ⊢ (i · i) ∈ ℂ |
8 | ax-1cn 11105 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | 4 | recni 11165 | . . . . . . 7 ⊢ (1 −ℝ 1) ∈ ℂ |
10 | 7, 8, 9 | addassi 11161 | . . . . . 6 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + (1 + (1 −ℝ 1))) |
11 | repncan3 40790 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (1 + (1 −ℝ 1)) = 1) | |
12 | 2, 2, 11 | mp2an 690 | . . . . . . 7 ⊢ (1 + (1 −ℝ 1)) = 1 |
13 | 12 | oveq2i 7364 | . . . . . 6 ⊢ ((i · i) + (1 + (1 −ℝ 1))) = ((i · i) + 1) |
14 | 10, 13 | eqtri 2764 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + 1) |
15 | ax-i2m1 11115 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
16 | 15 | oveq1i 7363 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = (0 + (1 −ℝ 1)) |
17 | 14, 16, 15 | 3eqtr3i 2772 | . . . 4 ⊢ (0 + (1 −ℝ 1)) = 0 |
18 | 17 | a1i 11 | . . 3 ⊢ (⊤ → (0 + (1 −ℝ 1)) = 0) |
19 | 1, 5, 18 | reladdrsub 40792 | . 2 ⊢ (⊤ → (1 −ℝ 1) = (0 −ℝ 0)) |
20 | 19 | mptru 1548 | 1 ⊢ (1 −ℝ 1) = (0 −ℝ 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 (class class class)co 7353 ℝcr 11046 0cc0 11047 1c1 11048 ici 11049 + caddc 11050 · cmul 11052 −ℝ cresub 40772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-addass 11112 ax-i2m1 11115 ax-1ne0 11116 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-ltxr 11190 df-resub 40773 |
This theorem is referenced by: sn-00idlem1 40805 sn-00idlem2 40806 remul02 40812 |
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