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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 resubidaddlid 42425. Uses ax-i2m1 11223. (Contributed by SN, 25-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| re1m1e0m0 | ⊢ (1 −ℝ 1) = (0 −ℝ 0) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0red 11264 | . . 3 ⊢ (⊤ → 0 ∈ ℝ) | |
| 2 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 3 | rersubcl 42408 | . . . . 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 11214 | . . . . . . . 8 ⊢ i ∈ ℂ | |
| 7 | 6, 6 | mulcli 11268 | . . . . . . 7 ⊢ (i · i) ∈ ℂ | 
| 8 | ax-1cn 11213 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
| 9 | 4 | recni 11275 | . . . . . . 7 ⊢ (1 −ℝ 1) ∈ ℂ | 
| 10 | 7, 8, 9 | addassi 11271 | . . . . . 6 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + (1 + (1 −ℝ 1))) | 
| 11 | repncan3 42413 | . . . . . . . 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 2765 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + 1) | 
| 15 | ax-i2m1 11223 | . . . . . 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 2773 | . . . 4 ⊢ (0 + (1 −ℝ 1)) = 0 | 
| 18 | 17 | a1i 11 | . . 3 ⊢ (⊤ → (0 + (1 −ℝ 1)) = 0) | 
| 19 | 1, 5, 18 | reladdrsub 42415 | . 2 ⊢ (⊤ → (1 −ℝ 1) = (0 −ℝ 0)) | 
| 20 | 19 | mptru 1547 | 1 ⊢ (1 −ℝ 1) = (0 −ℝ 0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 −ℝ cresub 42395 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-addass 11220 ax-i2m1 11223 ax-1ne0 11224 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-resub 42396 | 
| This theorem is referenced by: sn-00idlem1 42428 sn-00idlem2 42429 remul02 42435 | 
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