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Mirrors > Home > MPE Home > Th. List > comffval | Structured version Visualization version GIF version |
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffval.o | ⊢ 𝑂 = (compf‘𝐶) |
comfffval.b | ⊢ 𝐵 = (Base‘𝐶) |
comfffval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
comfffval.x | ⊢ · = (comp‘𝐶) |
comffval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
comffval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
comffval.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
comffval | ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffval.o | . . . 4 ⊢ 𝑂 = (compf‘𝐶) | |
2 | comfffval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | comfffval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | comfffval.x | . . . 4 ⊢ · = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | comfffval 16358 | . . 3 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓))) |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)))) |
7 | simprl 794 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑥 = 〈𝑋, 𝑌〉) | |
8 | 7 | fveq2d 6195 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = (2nd ‘〈𝑋, 𝑌〉)) |
9 | comffval.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | comffval.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | op2ndg 7181 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
12 | 9, 10, 11 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
14 | 8, 13 | eqtrd 2656 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = 𝑌) |
15 | simprr 796 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
16 | 14, 15 | oveq12d 6668 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑥)𝐻𝑧) = (𝑌𝐻𝑍)) |
17 | 7 | fveq2d 6195 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝐻‘〈𝑋, 𝑌〉)) |
18 | df-ov 6653 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
19 | 17, 18 | syl6eqr 2674 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝑋𝐻𝑌)) |
20 | 7, 15 | oveq12d 6668 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (〈𝑋, 𝑌〉 · 𝑍)) |
21 | 20 | oveqd 6667 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) |
22 | 16, 19, 21 | mpt2eq123dv 6717 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
23 | opelxpi 5148 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
24 | 9, 10, 23 | syl2anc 693 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
25 | comffval.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
26 | ovex 6678 | . . . 4 ⊢ (𝑌𝐻𝑍) ∈ V | |
27 | ovex 6678 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
28 | 26, 27 | mpt2ex 7247 | . . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) ∈ V |
29 | 28 | a1i 11 | . 2 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) ∈ V) |
30 | 6, 22, 24, 25, 29 | ovmpt2d 6788 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 2nd c2nd 7167 Basecbs 15857 Hom chom 15952 compcco 15953 compfccomf 16328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-comf 16332 |
This theorem is referenced by: comfval 16360 comffval2 16362 comffn 16365 |
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