Theorem addsproplem6 28366

Description: Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addspropord.2	⊢ (𝜑 → 𝑋 ∈ No)
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No)
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No)
addspropord.5	⊢ (𝜑 → 𝑌 <s 𝑍)
addsproplem6.6	⊢ (𝜑 → (bday‘𝑌) = (bday‘𝑍))

Assertion

Ref	Expression
addsproplem6	⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem addsproplem6

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addspropord.3	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No)
2		addspropord.4	. . . 4 ⊢ (𝜑 → 𝑍 ∈ No)
3		addsproplem6.6	. . . 4 ⊢ (𝜑 → (bday‘𝑌) = (bday‘𝑍))
4		addspropord.5	. . . 4 ⊢ (𝜑 → 𝑌 <s 𝑍)
5		nodense 28055	. . . 4 ⊢ (((𝑌 ∈ No ∧ 𝑍 ∈ No) ∧ ((bday‘𝑌) = (bday‘𝑍) ∧ 𝑌 <s 𝑍)) → ∃𝑚 ∈ No ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
6	1, 2, 3, 4, 5	syl22anc 852	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ No ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
7		addsproplem.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
8		addspropord.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ No)
9	7, 8, 1	addsproplem3 28363	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)})))
10	9	simp1d 1160	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ No)
12	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
13	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑋 ∈ No)
14		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ No)
15		unidm 4156	. . . . . . 7 ⊢ (((bday‘𝑋) +no (bday‘𝑚)) ∪ ((bday‘𝑋) +no (bday‘𝑚))) = ((bday‘𝑋) +no (bday‘𝑚))
16		simprr1 1240	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (bday‘𝑚) ∈ (bday‘𝑌))
17		bdayon 28144	. . . . . . . . . 10 ⊢ (bday‘𝑚) ∈ On
18		bdayon 28144	. . . . . . . . . 10 ⊢ (bday‘𝑌) ∈ On
19		bdayon 28144	. . . . . . . . . 10 ⊢ (bday‘𝑋) ∈ On
20		naddel2 8702	. . . . . . . . . 10 ⊢ (((bday‘𝑚) ∈ On ∧ (bday‘𝑌) ∈ On ∧ (bday‘𝑋) ∈ On) → ((bday‘𝑚) ∈ (bday‘𝑌) ↔ ((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌))))
21	17, 18, 19, 20	mp3an 1490	. . . . . . . . 9 ⊢ ((bday‘𝑚) ∈ (bday‘𝑌) ↔ ((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌)))
22	16, 21	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌)))
23		elun1 4180	. . . . . . . 8 ⊢ (((bday‘𝑋) +no (bday‘𝑚)) ∈ ((bday‘𝑋) +no (bday‘𝑌)) → ((bday‘𝑋) +no (bday‘𝑚)) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
24	22, 23	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((bday‘𝑋) +no (bday‘𝑚)) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
25	15, 24	eqeltrid 2864	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (((bday‘𝑋) +no (bday‘𝑚)) ∪ ((bday‘𝑋) +no (bday‘𝑚))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
26	12, 13, 14, 14, 25	addsproplem1 28361	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑚) ∈ No ∧ (𝑚 <s 𝑚 → (𝑚 +s 𝑋) <s (𝑚 +s 𝑋))))
27	26	simpld 500	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ No)
28		uncom 4157	. . . . . . . . . . . 12 ⊢ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) = (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌)))
29	28	eleq2i 2852	. . . . . . . . . . 11 ⊢ ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) ↔ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))))
30	29	imbi1i 352	. . . . . . . . . 10 ⊢ (((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
31	30	ralbii 3108	. . . . . . . . 9 ⊢ (∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
32	31	2ralbii 3137	. . . . . . . 8 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
33	7, 32	sylib 221	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
34	33, 8, 2	addsproplem3 28363	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑍)𝑔 = (𝑋 +s ℎ)})))
35	34	simp1d 1160	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ No)
36	35	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ No)
37	9	simp3d 1162	. . . . . 6 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
38	37	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
39		ovex 7454	. . . . . . 7 ⊢ (𝑋 +s 𝑌) ∈ V
40	39	snid 4619	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
41	40	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
42		oldbday 28293	. . . . . . . . . . 11 ⊢ (((bday‘𝑌) ∈ On ∧ 𝑚 ∈ No) → (𝑚 ∈ (O‘(bday‘𝑌)) ↔ (bday‘𝑚) ∈ (bday‘𝑌)))
43	18, 14, 42	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ (O‘(bday‘𝑌)) ↔ (bday‘𝑚) ∈ (bday‘𝑌)))
44	16, 43	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (O‘(bday‘𝑌)))
45		simprr2 1241	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑌 <s 𝑚)
46		elright 28244	. . . . . . . . 9 ⊢ (𝑚 ∈ (R‘𝑌) ↔ (𝑚 ∈ (O‘(bday‘𝑌)) ∧ 𝑌 <s 𝑚))
47	44, 45, 46	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (R‘𝑌))
48		eqid 2760	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)
49		oveq2 7429	. . . . . . . . 9 ⊢ (ℎ = 𝑚 → (𝑋 +s ℎ) = (𝑋 +s 𝑚))
50	49	rspceeqv 3598	. . . . . . . 8 ⊢ ((𝑚 ∈ (R‘𝑌) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
51	47, 48, 50	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
52		ovex 7454	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) ∈ V
53		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑔 = (𝑋 +s 𝑚) → (𝑔 = (𝑋 +s ℎ) ↔ (𝑋 +s 𝑚) = (𝑋 +s ℎ)))
54	53	rexbidv 3186	. . . . . . . 8 ⊢ (𝑔 = (𝑋 +s 𝑚) → (∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ)))
55	52, 54	elab 3632	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)} ↔ ∃ℎ ∈ (R‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
56	51, 55	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)})
57		elun2 4181	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)} → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
58	56, 57	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}))
59	38, 41, 58	sltssepcd 28164	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑚))
60	33	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
61	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑍 ∈ No)
62	60, 13, 61	addsproplem3 28363	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑍)𝑔 = (𝑋 +s ℎ)})))
63	62	simp2d 1161	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
64	3	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (bday‘𝑌) = (bday‘𝑍))
65	16, 64	eleqtrd 2862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (bday‘𝑚) ∈ (bday‘𝑍))
66		bdayon 28144	. . . . . . . . . . 11 ⊢ (bday‘𝑍) ∈ On
67		oldbday 28293	. . . . . . . . . . 11 ⊢ (((bday‘𝑍) ∈ On ∧ 𝑚 ∈ No) → (𝑚 ∈ (O‘(bday‘𝑍)) ↔ (bday‘𝑚) ∈ (bday‘𝑍)))
68	66, 14, 67	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ (O‘(bday‘𝑍)) ↔ (bday‘𝑚) ∈ (bday‘𝑍)))
69	65, 68	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (O‘(bday‘𝑍)))
70		simprr3 1242	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 <s 𝑍)
71		elleft 28243	. . . . . . . . 9 ⊢ (𝑚 ∈ (L‘𝑍) ↔ (𝑚 ∈ (O‘(bday‘𝑍)) ∧ 𝑚 <s 𝑍))
72	69, 70, 71	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ (L‘𝑍))
73		oveq2 7429	. . . . . . . . 9 ⊢ (𝑑 = 𝑚 → (𝑋 +s 𝑑) = (𝑋 +s 𝑚))
74	73	rspceeqv 3598	. . . . . . . 8 ⊢ ((𝑚 ∈ (L‘𝑍) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
75	72, 48, 74	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
76		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑐 = (𝑋 +s 𝑚) → (𝑐 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
77	76	rexbidv 3186	. . . . . . . 8 ⊢ (𝑐 = (𝑋 +s 𝑚) → (∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
78	52, 77	elab 3632	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ (L‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
79	75, 78	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)})
80		elun2 4181	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
81	79, 80	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
82		ovex 7454	. . . . . . 7 ⊢ (𝑋 +s 𝑍) ∈ V
83	82	snid 4619	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
84	83	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
85	63, 81, 84	sltssepcd 28164	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑍))
86	11, 27, 36, 59, 85	ltstrd 28126	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ ((bday‘𝑚) ∈ (bday‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
87	6, 86	rexlimddv 3169	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
88	1, 8	addscomd 28359	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
89	2, 8	addscomd 28359	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
90	87, 88, 89	3brtr4d 5136	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∪ cun 4149  {csn 4584   class class class wbr 5102  Oncon0 6362  ‘cfv 6539  (class class class)co 7421   +no cnadd 8678  Nocsur 28003   <s clts 28004  bdaycbday 28005   <<s cslts 28149  Ocold 28215  Lcleft 28217  Rcright 28218   +_s cadds 28351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-0s 28199  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec2 28341  df-adds 28352

This theorem is used by:  addsproplem7  28367