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Mirrors > Home > MPE Home > Th. List > cgrane3 | Structured version Visualization version GIF version |
Description: Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
Ref | Expression |
---|---|
iscgra.p | ⊢ 𝑃 = (Base‘𝐺) |
iscgra.i | ⊢ 𝐼 = (Itv‘𝐺) |
iscgra.k | ⊢ 𝐾 = (hlG‘𝐺) |
iscgra.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
iscgra.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
iscgra.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
iscgra.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
iscgra.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
iscgra.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
iscgra.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
cgrahl1.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Ref | Expression |
---|---|
cgrane3 | ⊢ (𝜑 → 𝐸 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscgra.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | iscgra.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | iscgra.k | . . . 4 ⊢ 𝐾 = (hlG‘𝐺) | |
4 | simpllr 774 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥 ∈ 𝑃) | |
5 | iscgra.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
6 | 5 | ad3antrrr 728 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐷 ∈ 𝑃) |
7 | iscgra.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
8 | 7 | ad3antrrr 728 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃) |
9 | iscgra.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
10 | 9 | ad3antrrr 728 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG) |
11 | simpr2 1191 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥(𝐾‘𝐸)𝐷) | |
12 | 1, 2, 3, 4, 6, 8, 10, 11 | hlne2 26391 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐷 ≠ 𝐸) |
13 | 12 | necomd 3071 | . 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐸 ≠ 𝐷) |
14 | cgrahl1.2 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | |
15 | iscgra.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
16 | iscgra.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
17 | iscgra.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
18 | iscgra.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
19 | 1, 2, 3, 9, 15, 16, 17, 5, 7, 18 | iscgra 26594 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))) |
20 | 14, 19 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) |
21 | 13, 20 | r19.29vva 3336 | 1 ⊢ (𝜑 → 𝐸 ≠ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 class class class wbr 5065 ‘cfv 6354 〈“cs3 14203 Basecbs 16482 TarskiGcstrkg 26215 Itvcitv 26221 cgrGccgrg 26295 hlGchlg 26385 cgrAccgra 26592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-hlg 26386 df-cgra 26593 |
This theorem is referenced by: cgracom 26607 cgratr 26608 cgraswaplr 26610 dfcgra2 26615 leagne3 26636 tgsas2 26641 tgasa1 26643 |
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