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Mirrors > Home > MPE Home > Th. List > ofsubeq0 | Structured version Visualization version GIF version |
Description: Function analogue of subeq0 11532. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
ofsubeq0 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹 ∘f − 𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1136 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → 𝐹:𝐴⟶ℂ) | |
2 | 1 | ffnd 6737 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → 𝐹 Fn 𝐴) |
3 | simp3 1137 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → 𝐺:𝐴⟶ℂ) | |
4 | 3 | ffnd 6737 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → 𝐺 Fn 𝐴) |
5 | simp1 1135 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → 𝐴 ∈ 𝑉) | |
6 | inidm 4234 | . . . . . 6 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | eqidd 2735 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
8 | eqidd 2735 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
9 | 2, 4, 5, 5, 6, 7, 8 | ofval 7707 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f − 𝐺)‘𝑥) = ((𝐹‘𝑥) − (𝐺‘𝑥))) |
10 | c0ex 11252 | . . . . . . 7 ⊢ 0 ∈ V | |
11 | 10 | fvconst2 7223 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
12 | 11 | adantl 481 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
13 | 9, 12 | eqeq12d 2750 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘f − 𝐺)‘𝑥) = ((𝐴 × {0})‘𝑥) ↔ ((𝐹‘𝑥) − (𝐺‘𝑥)) = 0)) |
14 | 1 | ffvelcdmda 7103 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℂ) |
15 | 3 | ffvelcdmda 7103 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ℂ) |
16 | 14, 15 | subeq0ad 11627 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝑥) − (𝐺‘𝑥)) = 0 ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
17 | 13, 16 | bitrd 279 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘f − 𝐺)‘𝑥) = ((𝐴 × {0})‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
18 | 17 | ralbidva 3173 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (∀𝑥 ∈ 𝐴 ((𝐹 ∘f − 𝐺)‘𝑥) = ((𝐴 × {0})‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
19 | 2, 4, 5, 5, 6 | offn 7709 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹 ∘f − 𝐺) Fn 𝐴) |
20 | 10 | fconst 6794 | . . . 4 ⊢ (𝐴 × {0}):𝐴⟶{0} |
21 | ffn 6736 | . . . 4 ⊢ ((𝐴 × {0}):𝐴⟶{0} → (𝐴 × {0}) Fn 𝐴) | |
22 | 20, 21 | ax-mp 5 | . . 3 ⊢ (𝐴 × {0}) Fn 𝐴 |
23 | eqfnfv 7050 | . . 3 ⊢ (((𝐹 ∘f − 𝐺) Fn 𝐴 ∧ (𝐴 × {0}) Fn 𝐴) → ((𝐹 ∘f − 𝐺) = (𝐴 × {0}) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘f − 𝐺)‘𝑥) = ((𝐴 × {0})‘𝑥))) | |
24 | 19, 22, 23 | sylancl 586 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹 ∘f − 𝐺) = (𝐴 × {0}) ↔ ∀𝑥 ∈ 𝐴 ((𝐹 ∘f − 𝐺)‘𝑥) = ((𝐴 × {0})‘𝑥))) |
25 | eqfnfv 7050 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
26 | 2, 4, 25 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
27 | 18, 24, 26 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹 ∘f − 𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 {csn 4630 × cxp 5686 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 ℂcc 11150 0cc0 11152 − cmin 11489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-sub 11491 |
This theorem is referenced by: psrridm 22000 dv11cn 26054 coeeulem 26277 plydiveu 26354 facth 26362 quotcan 26365 plyexmo 26369 mpaaeu 43138 |
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