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Mirrors > Home > MPE Home > Th. List > iscgrad | Structured version Visualization version GIF version |
Description: Sufficient conditions for angle congruence, deduction version. (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) |
iscgrad.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
iscgrad.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
iscgrad.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉) |
iscgrad.2 | ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝐷) |
iscgrad.3 | ⊢ (𝜑 → 𝑌(𝐾‘𝐸)𝐹) |
Ref | Expression |
---|---|
iscgrad | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscgrad.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
2 | iscgrad.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
3 | iscgrad.1 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉) | |
4 | iscgrad.2 | . . 3 ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝐷) | |
5 | iscgrad.3 | . . 3 ⊢ (𝜑 → 𝑌(𝐾‘𝐸)𝐹) | |
6 | id 22 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
7 | eqidd 2739 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝐸 = 𝐸) | |
8 | eqidd 2739 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑦 = 𝑦) | |
9 | 6, 7, 8 | s3eqd 14587 | . . . . . 6 ⊢ (𝑥 = 𝑋 → 〈“𝑥𝐸𝑦”〉 = 〈“𝑋𝐸𝑦”〉) |
10 | 9 | breq2d 5085 | . . . . 5 ⊢ (𝑥 = 𝑋 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉)) |
11 | breq1 5076 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(𝐾‘𝐸)𝐷 ↔ 𝑋(𝐾‘𝐸)𝐷)) | |
12 | 10, 11 | 3anbi12d 1436 | . . . 4 ⊢ (𝑥 = 𝑋 → ((〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹) ↔ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉 ∧ 𝑋(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))) |
13 | eqidd 2739 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑋 = 𝑋) | |
14 | eqidd 2739 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝐸 = 𝐸) | |
15 | id 22 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌) | |
16 | 13, 14, 15 | s3eqd 14587 | . . . . . 6 ⊢ (𝑦 = 𝑌 → 〈“𝑋𝐸𝑦”〉 = 〈“𝑋𝐸𝑌”〉) |
17 | 16 | breq2d 5085 | . . . . 5 ⊢ (𝑦 = 𝑌 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉)) |
18 | breq1 5076 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦(𝐾‘𝐸)𝐹 ↔ 𝑌(𝐾‘𝐸)𝐹)) | |
19 | 17, 18 | 3anbi13d 1437 | . . . 4 ⊢ (𝑦 = 𝑌 → ((〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉 ∧ 𝑋(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹) ↔ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉 ∧ 𝑋(𝐾‘𝐸)𝐷 ∧ 𝑌(𝐾‘𝐸)𝐹))) |
20 | 12, 19 | rspc2ev 3571 | . . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉 ∧ 𝑋(𝐾‘𝐸)𝐷 ∧ 𝑌(𝐾‘𝐸)𝐹)) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) |
21 | 1, 2, 3, 4, 5, 20 | syl113anc 1381 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) |
22 | iscgra.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
23 | iscgra.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
24 | iscgra.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
25 | iscgra.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
26 | iscgra.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
27 | iscgra.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
28 | iscgra.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
29 | iscgra.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
30 | iscgra.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
31 | iscgra.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
32 | 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 | iscgra 27180 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))) |
33 | 21, 32 | mpbird 256 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 class class class wbr 5073 ‘cfv 6426 〈“cs3 14565 Basecbs 16922 TarskiGcstrkg 26798 Itvcitv 26804 cgrGccgrg 26881 hlGchlg 26971 cgrAccgra 27178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-fzo 13393 df-hash 14055 df-word 14228 df-concat 14284 df-s1 14311 df-s2 14571 df-s3 14572 df-cgra 27179 |
This theorem is referenced by: cgrahl1 27187 cgrahl2 27188 cgraid 27190 cgrcgra 27192 dfcgra2 27201 sacgr 27202 tgsas2 27227 tgsas3 27228 tgasa1 27229 |
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