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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 2727 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝐸 = 𝐸) | |
8 | eqidd 2727 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑦 = 𝑦) | |
9 | 6, 7, 8 | s3eqd 14868 | . . . . . 6 ⊢ (𝑥 = 𝑋 → 〈“𝑥𝐸𝑦”〉 = 〈“𝑋𝐸𝑦”〉) |
10 | 9 | breq2d 5157 | . . . . 5 ⊢ (𝑥 = 𝑋 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉)) |
11 | breq1 5148 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(𝐾‘𝐸)𝐷 ↔ 𝑋(𝐾‘𝐸)𝐷)) | |
12 | 10, 11 | 3anbi12d 1434 | . . . 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 2727 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑋 = 𝑋) | |
14 | eqidd 2727 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝐸 = 𝐸) | |
15 | id 22 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌) | |
16 | 13, 14, 15 | s3eqd 14868 | . . . . . 6 ⊢ (𝑦 = 𝑌 → 〈“𝑋𝐸𝑦”〉 = 〈“𝑋𝐸𝑌”〉) |
17 | 16 | breq2d 5157 | . . . . 5 ⊢ (𝑦 = 𝑌 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉)) |
18 | breq1 5148 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦(𝐾‘𝐸)𝐹 ↔ 𝑌(𝐾‘𝐸)𝐹)) | |
19 | 17, 18 | 3anbi13d 1435 | . . . 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 3620 | . . 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 1379 | . 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 28733 | . 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 1084 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 class class class wbr 5145 ‘cfv 6546 〈“cs3 14846 Basecbs 17208 TarskiGcstrkg 28351 Itvcitv 28357 cgrGccgrg 28434 hlGchlg 28524 cgrAccgra 28731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-hash 14343 df-word 14518 df-concat 14574 df-s1 14599 df-s2 14852 df-s3 14853 df-cgra 28732 |
This theorem is referenced by: cgrahl1 28740 cgrahl2 28741 cgraid 28743 cgrcgra 28745 dfcgra2 28754 sacgr 28755 tgsas2 28780 tgsas3 28781 tgasa1 28782 |
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