Mathbox for Filip Cernatescu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > problem3 | Structured version Visualization version GIF version |
Description: Practice problem 3. Clues: eqcomi 2619 eqtri 2632 subaddrii 10249 recni 9931 4re 10974 3re 10971 1re 9918 df-4 10958 addcomi 10106. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
problem3.1 | ⊢ 𝐴 ∈ ℂ |
problem3.2 | ⊢ (𝐴 + 3) = 4 |
Ref | Expression |
---|---|
problem3 | ⊢ 𝐴 = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4re 10974 | . . . . . 6 ⊢ 4 ∈ ℝ | |
2 | 1 | recni 9931 | . . . . 5 ⊢ 4 ∈ ℂ |
3 | 3re 10971 | . . . . . 6 ⊢ 3 ∈ ℝ | |
4 | 3 | recni 9931 | . . . . 5 ⊢ 3 ∈ ℂ |
5 | 1re 9918 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 5 | recni 9931 | . . . . 5 ⊢ 1 ∈ ℂ |
7 | df-4 10958 | . . . . . 6 ⊢ 4 = (3 + 1) | |
8 | 7 | eqcomi 2619 | . . . . 5 ⊢ (3 + 1) = 4 |
9 | 2, 4, 6, 8 | subaddrii 10249 | . . . 4 ⊢ (4 − 3) = 1 |
10 | 9 | eqcomi 2619 | . . 3 ⊢ 1 = (4 − 3) |
11 | problem3.1 | . . . 4 ⊢ 𝐴 ∈ ℂ | |
12 | 4, 11 | addcomi 10106 | . . . . 5 ⊢ (3 + 𝐴) = (𝐴 + 3) |
13 | problem3.2 | . . . . 5 ⊢ (𝐴 + 3) = 4 | |
14 | 12, 13 | eqtri 2632 | . . . 4 ⊢ (3 + 𝐴) = 4 |
15 | 2, 4, 11, 14 | subaddrii 10249 | . . 3 ⊢ (4 − 3) = 𝐴 |
16 | 10, 15 | eqtri 2632 | . 2 ⊢ 1 = 𝐴 |
17 | 16 | eqcomi 2619 | 1 ⊢ 𝐴 = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 − cmin 10145 3c3 10948 4c4 10949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-2 10956 df-3 10957 df-4 10958 |
This theorem is referenced by: (None) |
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