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Mirrors > Home > MPE Home > Th. List > Mathboxes > cgrtr4d | Structured version Visualization version GIF version |
Description: Deduction form of axcgrtr 28152. (Contributed by Scott Fenton, 13-Oct-2013.) |
Ref | Expression |
---|---|
cgrtr4d.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
cgrtr4d.2 | ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) |
cgrtr4d.3 | ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) |
cgrtr4d.4 | ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) |
cgrtr4d.5 | ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) |
cgrtr4d.6 | ⊢ (𝜑 → 𝐸 ∈ (𝔼‘𝑁)) |
cgrtr4d.7 | ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁)) |
cgrtr4d.8 | ⊢ (𝜑 → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉) |
cgrtr4d.9 | ⊢ (𝜑 → 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) |
Ref | Expression |
---|---|
cgrtr4d | ⊢ (𝜑 → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgrtr4d.8 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉) | |
2 | cgrtr4d.9 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) | |
3 | cgrtr4d.1 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | cgrtr4d.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) | |
5 | cgrtr4d.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) | |
6 | cgrtr4d.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) | |
7 | cgrtr4d.5 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) | |
8 | cgrtr4d.6 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝔼‘𝑁)) | |
9 | cgrtr4d.7 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁)) | |
10 | axcgrtr 28152 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉)) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | syl133anc 1394 | . 2 ⊢ (𝜑 → ((〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐸, 𝐹〉) → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉)) |
12 | 1, 2, 11 | mp2and 698 | 1 ⊢ (𝜑 → 〈𝐶, 𝐷〉Cgr〈𝐸, 𝐹〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 〈cop 4632 class class class wbr 5146 ‘cfv 6539 ℕcn 12207 𝔼cee 28125 Cgrccgr 28127 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 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 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-seq 13962 df-sum 15628 df-ee 28128 df-cgr 28130 |
This theorem is referenced by: cgrtr4and 34895 cgrrflx 34896 segconeq 34919 |
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