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Mirrors > Home > MPE Home > Th. List > addneintr2d | Structured version Visualization version GIF version |
Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 10443. Consequence of addcan2d 10441. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
muld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcomd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
addneintr2d.4 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
addneintr2d | ⊢ (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addneintr2d.4 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | muld.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | addcomd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | addcand.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | 2, 3, 4 | addcan2d 10441 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) |
6 | 5 | necon3bid 2987 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) ≠ (𝐵 + 𝐶) ↔ 𝐴 ≠ 𝐵)) |
7 | 1, 6 | mpbird 247 | 1 ⊢ (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 ≠ wne 2943 (class class class)co 6792 ℂcc 10135 + caddc 10140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-ltxr 10280 |
This theorem is referenced by: modsumfzodifsn 12950 chordthmlem 24779 fperdvper 40647 |
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