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 2828 eqtri 2842 subaddrii 10967 recni 10647 4re 11713 3re 11709 1re 10633 df-4 11694 addcomi 10823. (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 11713 | . . . . . 6 ⊢ 4 ∈ ℝ | |
2 | 1 | recni 10647 | . . . . 5 ⊢ 4 ∈ ℂ |
3 | 3re 11709 | . . . . . 6 ⊢ 3 ∈ ℝ | |
4 | 3 | recni 10647 | . . . . 5 ⊢ 3 ∈ ℂ |
5 | 1re 10633 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 5 | recni 10647 | . . . . 5 ⊢ 1 ∈ ℂ |
7 | df-4 11694 | . . . . . 6 ⊢ 4 = (3 + 1) | |
8 | 7 | eqcomi 2828 | . . . . 5 ⊢ (3 + 1) = 4 |
9 | 2, 4, 6, 8 | subaddrii 10967 | . . . 4 ⊢ (4 − 3) = 1 |
10 | 9 | eqcomi 2828 | . . 3 ⊢ 1 = (4 − 3) |
11 | problem3.1 | . . . 4 ⊢ 𝐴 ∈ ℂ | |
12 | 4, 11 | addcomi 10823 | . . . . 5 ⊢ (3 + 𝐴) = (𝐴 + 3) |
13 | problem3.2 | . . . . 5 ⊢ (𝐴 + 3) = 4 | |
14 | 12, 13 | eqtri 2842 | . . . 4 ⊢ (3 + 𝐴) = 4 |
15 | 2, 4, 11, 14 | subaddrii 10967 | . . 3 ⊢ (4 − 3) = 𝐴 |
16 | 10, 15 | eqtri 2842 | . 2 ⊢ 1 = 𝐴 |
17 | 16 | eqcomi 2828 | 1 ⊢ 𝐴 = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∈ wcel 2108 (class class class)co 7148 ℂcc 10527 1c1 10530 + caddc 10532 − cmin 10862 3c3 11685 4c4 11686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 df-sub 10864 df-2 11692 df-3 11693 df-4 11694 |
This theorem is referenced by: (None) |
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