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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 39274. Uses ax-i2m1 10605. (Contributed by SN, 25-Dec-2023.) |
Ref | Expression |
---|---|
re1m1e0m0 | ⊢ (1 −ℝ 1) = (0 −ℝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10644 | . . 3 ⊢ (⊤ → 0 ∈ ℝ) | |
2 | 1re 10641 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rersubcl 39257 | . . . . 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 10596 | . . . . . . . 8 ⊢ i ∈ ℂ | |
7 | 6, 6 | mulcli 10648 | . . . . . . 7 ⊢ (i · i) ∈ ℂ |
8 | ax-1cn 10595 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
9 | 4 | recni 10655 | . . . . . . 7 ⊢ (1 −ℝ 1) ∈ ℂ |
10 | 7, 8, 9 | addassi 10651 | . . . . . 6 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + (1 + (1 −ℝ 1))) |
11 | repncan3 39262 | . . . . . . . 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 7167 | . . . . . 6 ⊢ ((i · i) + (1 + (1 −ℝ 1))) = ((i · i) + 1) |
14 | 10, 13 | eqtri 2844 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = ((i · i) + 1) |
15 | ax-i2m1 10605 | . . . . . 6 ⊢ ((i · i) + 1) = 0 | |
16 | 15 | oveq1i 7166 | . . . . 5 ⊢ (((i · i) + 1) + (1 −ℝ 1)) = (0 + (1 −ℝ 1)) |
17 | 14, 16, 15 | 3eqtr3i 2852 | . . . 4 ⊢ (0 + (1 −ℝ 1)) = 0 |
18 | 17 | a1i 11 | . . 3 ⊢ (⊤ → (0 + (1 −ℝ 1)) = 0) |
19 | 1, 5, 18 | reladdrsub 39264 | . 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 2114 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 −ℝ cresub 39244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-addass 10602 ax-i2m1 10605 ax-1ne0 10606 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-resub 39245 |
This theorem is referenced by: sn-00idlem1 39277 sn-00idlem2 39278 remul02 39284 |
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