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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 40038. Uses ax-i2m1 10780. (Contributed by SN, 25-Dec-2023.) |
Ref | Expression |
---|---|
re1m1e0m0 | ⊢ (1 −ℝ 1) = (0 −ℝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10819 | . . 3 ⊢ (⊤ → 0 ∈ ℝ) | |
2 | 1re 10816 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rersubcl 40021 | . . . . 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 10771 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | 6, 6 | mulcli 10823 | . . . . . . 7 ⊢ (i · i) ∈ ℂ |
8 | ax-1cn 10770 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | 4 | recni 10830 | . . . . . . 7 ⊢ (1 −ℝ 1) ∈ ℂ |
10 | 7, 8, 9 | addassi 10826 | . . . . . 6 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + (1 + (1 −ℝ 1))) |
11 | repncan3 40026 | . . . . . . . 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 7213 | . . . . . 6 ⊢ ((i · i) + (1 + (1 −ℝ 1))) = ((i · i) + 1) |
14 | 10, 13 | eqtri 2762 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + 1) |
15 | ax-i2m1 10780 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
16 | 15 | oveq1i 7212 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = (0 + (1 −ℝ 1)) |
17 | 14, 16, 15 | 3eqtr3i 2770 | . . . 4 ⊢ (0 + (1 −ℝ 1)) = 0 |
18 | 17 | a1i 11 | . . 3 ⊢ (⊤ → (0 + (1 −ℝ 1)) = 0) |
19 | 1, 5, 18 | reladdrsub 40028 | . 2 ⊢ (⊤ → (1 −ℝ 1) = (0 −ℝ 0)) |
20 | 19 | mptru 1550 | 1 ⊢ (1 −ℝ 1) = (0 −ℝ 0) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 (class class class)co 7202 ℝcr 10711 0cc0 10712 1c1 10713 ici 10714 + caddc 10715 · cmul 10717 −ℝ cresub 40008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-addass 10777 ax-i2m1 10780 ax-1ne0 10781 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-ltxr 10855 df-resub 40009 |
This theorem is referenced by: sn-00idlem1 40041 sn-00idlem2 40042 remul02 40048 |
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