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Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfioobdlem | Structured version Visualization version GIF version |
Description: 𝐺 actually extends 𝐹. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
cncfioobdlem.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
cncfioobdlem.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
cncfioobdlem.f | ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶𝑉) |
cncfioobdlem.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
cncfioobdlem.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) |
Ref | Expression |
---|---|
cncfioobdlem | ⊢ (𝜑 → (𝐺‘𝐶) = (𝐹‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfioobdlem.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))) |
3 | cncfioobdlem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 ∈ ℝ) |
5 | cncfioobdlem.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
6 | 3 | rexrd 11268 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
7 | cncfioobdlem.b | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
8 | 7 | rexrd 11268 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | elioo2 13369 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
10 | 6, 8, 9 | syl2anc 582 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
11 | 5, 10 | mpbid 231 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) |
12 | 11 | simp2d 1141 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐶) |
13 | 12 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 < 𝐶) |
14 | eqcom 2737 | . . . . . . . . 9 ⊢ (𝑥 = 𝐶 ↔ 𝐶 = 𝑥) | |
15 | 14 | biimpi 215 | . . . . . . . 8 ⊢ (𝑥 = 𝐶 → 𝐶 = 𝑥) |
16 | 15 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 = 𝑥) |
17 | 13, 16 | breqtrd 5173 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 < 𝑥) |
18 | 4, 17 | gtned 11353 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ≠ 𝐴) |
19 | 18 | neneqd 2943 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → ¬ 𝑥 = 𝐴) |
20 | 19 | iffalsed 4538 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) |
21 | simpr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶) | |
22 | 5 | elioored 44560 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
23 | 22 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ ℝ) |
24 | 21, 23 | eqeltrd 2831 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ℝ) |
25 | 11 | simp3d 1142 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 < 𝐵) |
26 | 25 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 < 𝐵) |
27 | 21, 26 | eqbrtrd 5169 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 < 𝐵) |
28 | 24, 27 | ltned 11354 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ≠ 𝐵) |
29 | 28 | neneqd 2943 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → ¬ 𝑥 = 𝐵) |
30 | 29 | iffalsed 4538 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥)) |
31 | 21 | fveq2d 6894 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝐹‘𝑥) = (𝐹‘𝐶)) |
32 | 20, 30, 31 | 3eqtrd 2774 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = (𝐹‘𝐶)) |
33 | ioossicc 13414 | . . 3 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
34 | 33, 5 | sselid 3979 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
35 | cncfioobdlem.f | . . 3 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶𝑉) | |
36 | 35, 5 | ffvelcdmd 7086 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑉) |
37 | 2, 32, 34, 36 | fvmptd 7004 | 1 ⊢ (𝜑 → (𝐺‘𝐶) = (𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 ℝ*cxr 11251 < clt 11252 (,)cioo 13328 [,]cicc 13331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioo 13332 df-icc 13335 |
This theorem is referenced by: cncfioobd 44911 |
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