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Mirrors > Home > MPE Home > Th. List > Mathboxes > naddwordnexlem0 | Structured version Visualization version GIF version |
Description: When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, (ω ·o suc 𝐶) lies between 𝐴 and 𝐵. (Contributed by RP, 14-Feb-2025.) |
Ref | Expression |
---|---|
naddwordnex.a | ⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀)) |
naddwordnex.b | ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁)) |
naddwordnex.c | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
naddwordnex.d | ⊢ (𝜑 → 𝐷 ∈ On) |
naddwordnex.m | ⊢ (𝜑 → 𝑀 ∈ ω) |
naddwordnex.n | ⊢ (𝜑 → 𝑁 ∈ 𝑀) |
Ref | Expression |
---|---|
naddwordnexlem0 | ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 9647 | . . . . . 6 ⊢ ω ∈ On | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → ω ∈ On) |
3 | naddwordnex.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ On) | |
4 | naddwordnex.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
5 | onelon 6389 | . . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On) | |
6 | 3, 4, 5 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On) |
7 | omcl 8542 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On) | |
8 | 2, 6, 7 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On) |
9 | 2, 8 | jca 511 | . . . 4 ⊢ (𝜑 → (ω ∈ On ∧ (ω ·o 𝐶) ∈ On)) |
10 | naddwordnex.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ω) | |
11 | oaordi 8552 | . . . 4 ⊢ ((ω ∈ On ∧ (ω ·o 𝐶) ∈ On) → (𝑀 ∈ ω → ((ω ·o 𝐶) +o 𝑀) ∈ ((ω ·o 𝐶) +o ω))) | |
12 | 9, 10, 11 | sylc 65 | . . 3 ⊢ (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ ((ω ·o 𝐶) +o ω)) |
13 | naddwordnex.a | . . 3 ⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀)) | |
14 | omsuc 8532 | . . . 4 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω)) | |
15 | 2, 6, 14 | syl2anc 583 | . . 3 ⊢ (𝜑 → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω)) |
16 | 12, 13, 15 | 3eltr4d 2847 | . 2 ⊢ (𝜑 → 𝐴 ∈ (ω ·o suc 𝐶)) |
17 | onsuc 7803 | . . . . . . 7 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On) | |
18 | 6, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → suc 𝐶 ∈ On) |
19 | 18, 3, 2 | 3jca 1127 | . . . . 5 ⊢ (𝜑 → (suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On)) |
20 | onsucss 42479 | . . . . . 6 ⊢ (𝐷 ∈ On → (𝐶 ∈ 𝐷 → suc 𝐶 ⊆ 𝐷)) | |
21 | 3, 4, 20 | sylc 65 | . . . . 5 ⊢ (𝜑 → suc 𝐶 ⊆ 𝐷) |
22 | omwordi 8577 | . . . . 5 ⊢ ((suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On) → (suc 𝐶 ⊆ 𝐷 → (ω ·o suc 𝐶) ⊆ (ω ·o 𝐷))) | |
23 | 19, 21, 22 | sylc 65 | . . . 4 ⊢ (𝜑 → (ω ·o suc 𝐶) ⊆ (ω ·o 𝐷)) |
24 | omcl 8542 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On) | |
25 | 2, 3, 24 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (ω ·o 𝐷) ∈ On) |
26 | naddwordnex.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑀) | |
27 | 26, 10 | jca 511 | . . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω)) |
28 | ontr1 6410 | . . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω)) | |
29 | 2, 27, 28 | sylc 65 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω) |
30 | nnon 7865 | . . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On) | |
31 | 29, 30 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ On) |
32 | oaword1 8558 | . . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁)) | |
33 | 25, 31, 32 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁)) |
34 | 23, 33 | sstrd 3992 | . . 3 ⊢ (𝜑 → (ω ·o suc 𝐶) ⊆ ((ω ·o 𝐷) +o 𝑁)) |
35 | naddwordnex.b | . . 3 ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁)) | |
36 | 34, 35 | sseqtrrd 4023 | . 2 ⊢ (𝜑 → (ω ·o suc 𝐶) ⊆ 𝐵) |
37 | 16, 36 | jca 511 | 1 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 Oncon0 6364 suc csuc 6366 (class class class)co 7412 ωcom 7859 +o coa 8469 ·o comu 8470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 ax-inf2 9642 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 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-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-oadd 8476 df-omul 8477 |
This theorem is referenced by: naddwordnexlem1 42611 naddwordnexlem2 42612 naddwordnexlem3 42613 |
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