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Mirrors > Home > MPE Home > Th. List > Mathboxes > no2indslem | Structured version Visualization version GIF version |
Description: Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.) |
Ref | Expression |
---|---|
no2indslem.a | ⊢ 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} |
no2indslem.b | ⊢ 𝑆 = {〈𝑐, 𝑑〉 ∣ (𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ (((1st ‘𝑐)𝑅(1st ‘𝑑) ∨ (1st ‘𝑐) = (1st ‘𝑑)) ∧ ((2nd ‘𝑐)𝑅(2nd ‘𝑑) ∨ (2nd ‘𝑐) = (2nd ‘𝑑)) ∧ 𝑐 ≠ 𝑑))} |
no2indslem.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) |
no2indslem.2 | ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒)) |
no2indslem.3 | ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒)) |
no2indslem.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
no2indslem.5 | ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂)) |
no2indslem.i | ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑)) |
Ref | Expression |
---|---|
no2indslem | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | no2indslem.b | . 2 ⊢ 𝑆 = {〈𝑐, 𝑑〉 ∣ (𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ (((1st ‘𝑐)𝑅(1st ‘𝑑) ∨ (1st ‘𝑐) = (1st ‘𝑑)) ∧ ((2nd ‘𝑐)𝑅(2nd ‘𝑑) ∨ (2nd ‘𝑐) = (2nd ‘𝑑)) ∧ 𝑐 ≠ 𝑑))} | |
2 | no2indslem.a | . . 3 ⊢ 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} | |
3 | 2 | lrrecfr 33682 | . 2 ⊢ 𝑅 Fr No |
4 | 2 | lrrecpo 33680 | . 2 ⊢ 𝑅 Po No |
5 | 2 | lrrecse 33681 | . 2 ⊢ 𝑅 Se No |
6 | no2indslem.1 | . 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) | |
7 | no2indslem.2 | . 2 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒)) | |
8 | no2indslem.3 | . 2 ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒)) | |
9 | no2indslem.4 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
10 | no2indslem.5 | . 2 ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂)) | |
11 | 2 | lrrecpred 33683 | . . . . . 6 ⊢ (𝑥 ∈ No → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥))) |
12 | 11 | adantr 484 | . . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥))) |
13 | 2 | lrrecpred 33683 | . . . . . . 7 ⊢ (𝑦 ∈ No → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦))) |
14 | 13 | adantl 485 | . . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦))) |
15 | 14 | raleqdv 3329 | . . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒)) |
16 | 12, 15 | raleqbidv 3319 | . . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒)) |
17 | 12 | raleqdv 3329 | . . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓)) |
18 | 14 | raleqdv 3329 | . . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃)) |
19 | 16, 17, 18 | 3anbi123d 1433 | . . 3 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ Pred (𝑅, No , 𝑥)∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ∧ ∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ∧ ∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃) ↔ (∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃))) |
20 | no2indslem.i | . . 3 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑)) | |
21 | 19, 20 | sylbid 243 | . 2 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ Pred (𝑅, No , 𝑥)∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ∧ ∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ∧ ∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃) → 𝜑)) |
22 | 1, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 21 | xpord2ind 33361 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∀wral 3070 ∪ cun 3858 class class class wbr 5036 {copab 5098 × cxp 5526 Predcpred 6130 ‘cfv 6340 1st c1st 7697 2nd c2nd 7698 No csur 33440 L cleft 33623 R cright 33624 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-1o 8118 df-2o 8119 df-no 33443 df-slt 33444 df-bday 33445 df-sslt 33573 df-scut 33575 df-made 33625 df-old 33626 df-left 33628 df-right 33629 |
This theorem is referenced by: no2inds 33694 |
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