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Mirrors > Home > MPE Home > Th. List > isinagd | Structured version Visualization version GIF version |
Description: Sufficient conditions for in-angle relation, deduction version. (Contributed by Thierry Arnoux, 20-Oct-2020.) |
Ref | Expression |
---|---|
isinag.p | ⊢ 𝑃 = (Base‘𝐺) |
isinag.i | ⊢ 𝐼 = (Itv‘𝐺) |
isinag.k | ⊢ 𝐾 = (hlG‘𝐺) |
isinag.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
isinag.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
isinag.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
isinag.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
isinagd.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
isinagd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
isinagd.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
isinagd.2 | ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
isinagd.3 | ⊢ (𝜑 → 𝑋 ≠ 𝐵) |
isinagd.4 | ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐼𝐶)) |
isinagd.5 | ⊢ (𝜑 → (𝑌 = 𝐵 ∨ 𝑌(𝐾‘𝐵)𝑋)) |
Ref | Expression |
---|---|
isinagd | ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinagd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | isinagd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) | |
3 | isinagd.3 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝐵) | |
4 | 1, 2, 3 | 3jca 1124 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵)) |
5 | isinagd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
6 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → 𝑥 = 𝑌) | |
7 | eqidd 2822 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝐴𝐼𝐶) = (𝐴𝐼𝐶)) | |
8 | 6, 7 | eleq12d 2907 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 ∈ (𝐴𝐼𝐶) ↔ 𝑌 ∈ (𝐴𝐼𝐶))) |
9 | eqidd 2822 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → 𝐵 = 𝐵) | |
10 | 6, 9 | eqeq12d 2837 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 = 𝐵 ↔ 𝑌 = 𝐵)) |
11 | 6 | breq1d 5068 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥(𝐾‘𝐵)𝑋 ↔ 𝑌(𝐾‘𝐵)𝑋)) |
12 | 10, 11 | orbi12d 915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → ((𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋) ↔ (𝑌 = 𝐵 ∨ 𝑌(𝐾‘𝐵)𝑋))) |
13 | 8, 12 | anbi12d 632 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → ((𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)) ↔ (𝑌 ∈ (𝐴𝐼𝐶) ∧ (𝑌 = 𝐵 ∨ 𝑌(𝐾‘𝐵)𝑋)))) |
14 | isinagd.4 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐼𝐶)) | |
15 | isinagd.5 | . . . . 5 ⊢ (𝜑 → (𝑌 = 𝐵 ∨ 𝑌(𝐾‘𝐵)𝑋)) | |
16 | 14, 15 | jca 514 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝐴𝐼𝐶) ∧ (𝑌 = 𝐵 ∨ 𝑌(𝐾‘𝐵)𝑋))) |
17 | 5, 13, 16 | rspcedvd 3625 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))) |
18 | 4, 17 | jca 514 | . 2 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
19 | isinag.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
20 | isinag.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
21 | isinag.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
22 | isinag.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
23 | isinag.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
24 | isinag.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
25 | isinag.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
26 | isinagd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
27 | 19, 20, 21, 22, 23, 24, 25, 26 | isinag 26618 | . 2 ⊢ (𝜑 → (𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉 ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))) |
28 | 18, 27 | mpbird 259 | 1 ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 〈“cs3 14198 Basecbs 16477 Itvcitv 26216 hlGchlg 26380 inAcinag 26615 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-inag 26617 |
This theorem is referenced by: inagflat 26620 |
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