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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 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 ∈ ℝ) |
5 | cncfioobdlem.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
6 | 3 | rexrd 11263 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
7 | cncfioobdlem.b | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
8 | 7 | rexrd 11263 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | elioo2 13364 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
10 | 6, 8, 9 | syl2anc 584 | . . . . . . . . . 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 1143 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐶) |
13 | 12 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 < 𝐶) |
14 | eqcom 2739 | . . . . . . . . 9 ⊢ (𝑥 = 𝐶 ↔ 𝐶 = 𝑥) | |
15 | 14 | biimpi 215 | . . . . . . . 8 ⊢ (𝑥 = 𝐶 → 𝐶 = 𝑥) |
16 | 15 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 = 𝑥) |
17 | 13, 16 | breqtrd 5174 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 < 𝑥) |
18 | 4, 17 | gtned 11348 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ≠ 𝐴) |
19 | 18 | neneqd 2945 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → ¬ 𝑥 = 𝐴) |
20 | 19 | iffalsed 4539 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) |
21 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶) | |
22 | 5 | elioored 44252 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
23 | 22 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ ℝ) |
24 | 21, 23 | eqeltrd 2833 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ℝ) |
25 | 11 | simp3d 1144 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 < 𝐵) |
26 | 25 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 < 𝐵) |
27 | 21, 26 | eqbrtrd 5170 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 < 𝐵) |
28 | 24, 27 | ltned 11349 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ≠ 𝐵) |
29 | 28 | neneqd 2945 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → ¬ 𝑥 = 𝐵) |
30 | 29 | iffalsed 4539 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥)) |
31 | 21 | fveq2d 6895 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝐹‘𝑥) = (𝐹‘𝐶)) |
32 | 20, 30, 31 | 3eqtrd 2776 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = (𝐹‘𝐶)) |
33 | ioossicc 13409 | . . 3 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
34 | 33, 5 | sselid 3980 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
35 | cncfioobdlem.f | . . 3 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶𝑉) | |
36 | 35, 5 | ffvelcdmd 7087 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑉) |
37 | 2, 32, 34, 36 | fvmptd 7005 | 1 ⊢ (𝜑 → (𝐺‘𝐶) = (𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℝcr 11108 ℝ*cxr 11246 < clt 11247 (,)cioo 13323 [,]cicc 13326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-ioo 13327 df-icc 13330 |
This theorem is referenced by: cncfioobd 44603 |
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