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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofsubid | Structured version Visualization version GIF version |
Description: Function analogue of subid 10742. (Contributed by Steve Rodriguez, 5-Nov-2015.) |
Ref | Expression |
---|---|
ofsubid | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∘𝑓 − 𝐹) = (𝐴 × {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → 𝐴 ∈ 𝑉) | |
2 | ffn 6374 | . . 3 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴) | |
3 | 2 | adantl 482 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → 𝐹 Fn 𝐴) |
4 | c0ex 10470 | . . . 4 ⊢ 0 ∈ V | |
5 | 4 | fconst 6425 | . . 3 ⊢ (𝐴 × {0}):𝐴⟶{0} |
6 | ffn 6374 | . . 3 ⊢ ((𝐴 × {0}):𝐴⟶{0} → (𝐴 × {0}) Fn 𝐴) | |
7 | 5, 6 | mp1i 13 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → (𝐴 × {0}) Fn 𝐴) |
8 | eqidd 2794 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
9 | ffvelrn 6705 | . . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℂ) | |
10 | 9 | subidd 10822 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = 0) |
11 | 10 | adantll 710 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = 0) |
12 | 4 | fvconst2 6824 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
13 | 12 | adantl 482 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
14 | 11, 13 | eqtr4d 2832 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) − (𝐹‘𝑥)) = ((𝐴 × {0})‘𝑥)) |
15 | 1, 3, 3, 7, 8, 8, 14 | offveq 7279 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∘𝑓 − 𝐹) = (𝐴 × {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 {csn 4466 × cxp 5433 Fn wfn 6212 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 ∘𝑓 cof 7256 ℂcc 10370 0cc0 10372 − cmin 10706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-ltxr 10515 df-sub 10708 |
This theorem is referenced by: expgrowth 40157 |
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