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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 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉) | |
4 | iscgrad.2 | . . . 4 ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝐷) | |
5 | iscgrad.3 | . . . 4 ⊢ (𝜑 → 𝑌(𝐾‘𝐸)𝐹) | |
6 | 3, 4, 5 | 3jca 1235 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉 ∧ 𝑋(𝐾‘𝐸)𝐷 ∧ 𝑌(𝐾‘𝐸)𝐹)) |
7 | id 22 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
8 | eqidd 2611 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝐸 = 𝐸) | |
9 | eqidd 2611 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑦 = 𝑦) | |
10 | 7, 8, 9 | s3eqd 13460 | . . . . . 6 ⊢ (𝑥 = 𝑋 → 〈“𝑥𝐸𝑦”〉 = 〈“𝑋𝐸𝑦”〉) |
11 | 10 | breq2d 4595 | . . . . 5 ⊢ (𝑥 = 𝑋 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉)) |
12 | breq1 4586 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥(𝐾‘𝐸)𝐷 ↔ 𝑋(𝐾‘𝐸)𝐷)) | |
13 | 11, 12 | 3anbi12d 1392 | . . . 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))) |
14 | eqidd 2611 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑋 = 𝑋) | |
15 | eqidd 2611 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝐸 = 𝐸) | |
16 | id 22 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌) | |
17 | 14, 15, 16 | s3eqd 13460 | . . . . . 6 ⊢ (𝑦 = 𝑌 → 〈“𝑋𝐸𝑦”〉 = 〈“𝑋𝐸𝑌”〉) |
18 | 17 | breq2d 4595 | . . . . 5 ⊢ (𝑦 = 𝑌 → (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑦”〉 ↔ 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑋𝐸𝑌”〉)) |
19 | breq1 4586 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦(𝐾‘𝐸)𝐹 ↔ 𝑌(𝐾‘𝐸)𝐹)) | |
20 | 18, 19 | 3anbi13d 1393 | . . . 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))) |
21 | 13, 20 | rspc2ev 3295 | . . 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)) |
22 | 1, 2, 6, 21 | syl3anc 1318 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) |
23 | iscgra.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
24 | iscgra.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
25 | iscgra.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
26 | iscgra.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
27 | iscgra.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
28 | iscgra.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
29 | iscgra.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
30 | iscgra.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
31 | iscgra.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
32 | iscgra.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
33 | 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 | iscgra 25501 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))) |
34 | 22, 33 | mpbird 246 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 〈“cs3 13438 Basecbs 15695 TarskiGcstrkg 25129 Itvcitv 25135 cgrGccgrg 25205 hlGchlg 25295 cgrAccgra 25499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-cgra 25500 |
This theorem is referenced by: cgrahl1 25508 cgrahl2 25509 cgraid 25511 cgrcgra 25513 dfcgra2 25521 sacgr 25522 tgsas2 25537 tgsas3 25538 tgasa1 25539 tgsss1 25541 |
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