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Mathbox for metakunt |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt23 | Structured version Visualization version GIF version |
Description: B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
metakunt23.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
metakunt23.2 | ⊢ (𝜑 → 𝐼 ∈ ℕ) |
metakunt23.3 | ⊢ (𝜑 → 𝐼 ≤ 𝑀) |
metakunt23.4 | ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) |
metakunt23.5 | ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) |
metakunt23.6 | ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) |
metakunt23.7 | ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
Ref | Expression |
---|---|
metakunt23 | ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metakunt23.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | 1 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑀 ∈ ℕ) |
3 | metakunt23.2 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
4 | 3 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝐼 ∈ ℕ) |
5 | metakunt23.3 | . . . 4 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
6 | 5 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝐼 ≤ 𝑀) |
7 | metakunt23.4 | . . 3 ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) | |
8 | metakunt23.5 | . . 3 ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) | |
9 | metakunt23.6 | . . 3 ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) | |
10 | metakunt23.7 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) | |
11 | 10 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑋 ∈ (1...𝑀)) |
12 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑋 = 𝑀) | |
13 | 2, 4, 6, 7, 8, 9, 11, 12 | metakunt20 41731 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) |
14 | 1 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝑀 ∈ ℕ) |
15 | 3 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℕ) |
16 | 5 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀) |
17 | 10 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀)) |
18 | simplr 767 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀) | |
19 | simpr 483 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝑋 < 𝐼) | |
20 | 14, 15, 16, 7, 8, 9, 17, 18, 19 | metakunt21 41732 | . . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) |
21 | 1 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝑀 ∈ ℕ) |
22 | 3 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝐼 ∈ ℕ) |
23 | 5 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀) |
24 | 10 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀)) |
25 | simplr 767 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀) | |
26 | simpr 483 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 < 𝐼) | |
27 | 21, 22, 23, 7, 8, 9, 24, 25, 26 | metakunt22 41733 | . . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) |
28 | 20, 27 | pm2.61dan 811 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) |
29 | 13, 28 | pm2.61dan 811 | 1 ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3938 ifcif 4524 {csn 4624 ⟨cop 4630 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6542 (class class class)co 7415 1c1 11137 + caddc 11139 < clt 11276 ≤ cle 11277 − cmin 11472 ℕcn 12240 ...cfz 13514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 |
This theorem is referenced by: metakunt25 41736 |
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