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Mirrors > Home > MPE Home > Th. List > isleagd | Structured version Visualization version GIF version |
Description: Sufficient condition for "less than" angle relation, deduction version (Contributed by Thierry Arnoux, 12-Oct-2020.) |
Ref | Expression |
---|---|
isleag.p | ⊢ 𝑃 = (Base‘𝐺) |
isleag.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
isleag.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
isleag.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
isleag.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
isleag.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
isleag.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
isleag.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
isleagd.s | ⊢ ≤ = (≤∠‘𝐺) |
isleagd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
isleagd.1 | ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉) |
isleagd.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉) |
Ref | Expression |
---|---|
isleagd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ≤ 〈“𝐷𝐸𝐹”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isleagd.s | . . . 4 ⊢ ≤ = (≤∠‘𝐺) | |
2 | 1 | eqcomi 2830 | . . 3 ⊢ (≤∠‘𝐺) = ≤ |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (≤∠‘𝐺) = ≤ ) |
4 | isleagd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
5 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
6 | 5 | breq1d 5076 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ 𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉)) |
7 | eqidd 2822 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐷 = 𝐷) | |
8 | eqidd 2822 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐸 = 𝐸) | |
9 | 7, 8, 5 | s3eqd 14226 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈“𝐷𝐸𝑥”〉 = 〈“𝐷𝐸𝑋”〉) |
10 | 9 | breq2d 5078 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉)) |
11 | 6, 10 | anbi12d 632 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉) ↔ (𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉))) |
12 | isleagd.1 | . . . . 5 ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉) | |
13 | isleagd.2 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉) | |
14 | 12, 13 | jca 514 | . . . 4 ⊢ (𝜑 → (𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉)) |
15 | 4, 11, 14 | rspcedvd 3626 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) |
16 | isleag.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
17 | isleag.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
18 | isleag.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
19 | isleag.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
20 | isleag.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
21 | isleag.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
22 | isleag.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
23 | isleag.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
24 | 16, 17, 18, 19, 20, 21, 22, 23 | isleag 26633 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉))) |
25 | 15, 24 | mpbird 259 | . 2 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) |
26 | 3, 25 | breqdi 5081 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ≤ 〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 〈“cs3 14204 Basecbs 16483 TarskiGcstrkg 26216 cgrAccgra 26593 inAcinag 26621 ≤∠cleag 26622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-leag 26632 |
This theorem is referenced by: (None) |
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