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Mirrors > Home > MPE Home > Th. List > Mathboxes > congid | Structured version Visualization version GIF version |
Description: Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
Ref | Expression |
---|---|
congid | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ (𝐵 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvds0 15969 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 0) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ 0) |
3 | zcn 12312 | . . . 4 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℂ) |
5 | 4 | subidd 11308 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 − 𝐵) = 0) |
6 | 2, 5 | breqtrrd 5102 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ (𝐵 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7268 ℂcc 10857 0cc0 10859 − cmin 11193 ℤcz 12307 ∥ cdvds 15951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-po 5499 df-so 5500 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-ltxr 11002 df-sub 11195 df-neg 11196 df-z 12308 df-dvds 15952 |
This theorem is referenced by: mzpcong 40780 acongid 40783 acongeq 40791 jm2.18 40796 jm2.25 40807 jm2.15nn0 40811 jm2.16nn0 40812 jm2.27c 40815 |
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