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Mirrors > Home > MPE Home > Th. List > pmtr3ncomlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for pmtr3ncom 18839. (Contributed by AV, 17-Mar-2018.) |
Ref | Expression |
---|---|
pmtr3ncom.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
pmtr3ncom.f | ⊢ 𝐹 = (𝑇‘{𝑋, 𝑌}) |
pmtr3ncom.g | ⊢ 𝐺 = (𝑇‘{𝑌, 𝑍}) |
Ref | Expression |
---|---|
pmtr3ncomlem2 | ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑍 ∈ 𝐷) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (𝐺 ∘ 𝐹) ≠ (𝐹 ∘ 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtr3ncom.t | . . 3 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
2 | pmtr3ncom.f | . . 3 ⊢ 𝐹 = (𝑇‘{𝑋, 𝑌}) | |
3 | pmtr3ncom.g | . . 3 ⊢ 𝐺 = (𝑇‘{𝑌, 𝑍}) | |
4 | 1, 2, 3 | pmtr3ncomlem1 18837 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑍 ∈ 𝐷) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → ((𝐺 ∘ 𝐹)‘𝑋) ≠ ((𝐹 ∘ 𝐺)‘𝑋)) |
5 | fveq1 6705 | . . 3 ⊢ ((𝐺 ∘ 𝐹) = (𝐹 ∘ 𝐺) → ((𝐺 ∘ 𝐹)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑋)) | |
6 | 5 | necon3i 2967 | . 2 ⊢ (((𝐺 ∘ 𝐹)‘𝑋) ≠ ((𝐹 ∘ 𝐺)‘𝑋) → (𝐺 ∘ 𝐹) ≠ (𝐹 ∘ 𝐺)) |
7 | 4, 6 | syl 17 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑍 ∈ 𝐷) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (𝐺 ∘ 𝐹) ≠ (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 {cpr 4533 ∘ ccom 5544 ‘cfv 6369 pmTrspcpmtr 18805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-om 7634 df-1o 8191 df-2o 8192 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pmtr 18806 |
This theorem is referenced by: pmtr3ncom 18839 |
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