| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2746 | . . 3 ⊢ (≤∠‘𝐺) = ≤ |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (≤∠‘𝐺) = ≤ ) |
| 4 | isleagd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 5 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
| 6 | 5 | breq1d 5153 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ 𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉)) |
| 7 | eqidd 2738 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐷 = 𝐷) | |
| 8 | eqidd 2738 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐸 = 𝐸) | |
| 9 | 7, 8, 5 | s3eqd 14903 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 〈“𝐷𝐸𝑥”〉 = 〈“𝐷𝐸𝑋”〉) |
| 10 | 9 | breq2d 5155 | . . . . 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 511 | . . . 4 ⊢ (𝜑 → (𝑋(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑋”〉)) |
| 15 | 4, 11, 14 | rspcedvd 3624 | . . 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 28855 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉))) |
| 25 | 15, 24 | mpbird 257 | . 2 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) |
| 26 | 3, 25 | breqdi 5158 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ≤ 〈“𝐷𝐸𝐹”〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 ‘cfv 6561 〈“cs3 14881 Basecbs 17247 TarskiGcstrkg 28435 cgrAccgra 28815 inAcinag 28843 ≤∠cleag 28844 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-leag 28854 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |