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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 2819 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝐸 = 𝐸) | |
8 | eqidd 2819 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑦 = 𝑦) | |
9 | 6, 7, 8 | s3eqd 14214 | . . . . . 6 ⊢ (𝑥 = 𝑋 → 〈“𝑥𝐸𝑦”〉 = 〈“𝑋𝐸𝑦”〉) |
10 | 9 | breq2d 5069 | . . . . 5 ⊢ (𝑥 = 𝑋 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉)) |
11 | breq1 5060 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(𝐾‘𝐸)𝐷 ↔ 𝑋(𝐾‘𝐸)𝐷)) | |
12 | 10, 11 | 3anbi12d 1428 | . . . 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 2819 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑋 = 𝑋) | |
14 | eqidd 2819 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝐸 = 𝐸) | |
15 | id 22 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌) | |
16 | 13, 14, 15 | s3eqd 14214 | . . . . . 6 ⊢ (𝑦 = 𝑌 → 〈“𝑋𝐸𝑦”〉 = 〈“𝑋𝐸𝑌”〉) |
17 | 16 | breq2d 5069 | . . . . 5 ⊢ (𝑦 = 𝑌 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉)) |
18 | breq1 5060 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦(𝐾‘𝐸)𝐹 ↔ 𝑌(𝐾‘𝐸)𝐹)) | |
19 | 17, 18 | 3anbi13d 1429 | . . . 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 3632 | . . 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 1374 | . 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 26522 | . 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 258 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 class class class wbr 5057 ‘cfv 6348 〈“cs3 14192 Basecbs 16471 TarskiGcstrkg 26143 Itvcitv 26149 cgrGccgrg 26223 hlGchlg 26313 cgrAccgra 26520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-cgra 26521 |
This theorem is referenced by: cgrahl1 26529 cgrahl2 26530 cgraid 26532 cgrcgra 26534 dfcgra2 26543 sacgr 26544 tgsas2 26569 tgsas3 26570 tgasa1 26571 |
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